The final size of the C`-free process Michael E. Picollelli∗ August 30, 2013

Abstract We consider the following random graph process: fix an integer ` ≥ 4, and starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of the cycle C` . Using the differential equations method, we show that, with probability tending to 1 as n → ∞, the final graph produced by this process has maximum degree O((n log n)1/(`−1) ) and, consequently, size O(n`/(`−1) (log n)1/(`−1) ). These results are sharp up to the hidden constants, improving upon previous bounds due to Osthus and Taraz.

1

Introduction

The H-free process, for a fixed graph H, is the random graph process which begins with a graph G(0) on n isolated vertices. The graph G(i) is then formed by adding an edge ei selected uniformly at random from the pairs of vertices which neither form edges of G(i − 1) nor create a copy of H in the graph G(i − 1) + ei . The process terminates with a maximal H-free graph G(MH ) with a random number MH of edges, and minimum and maximum degrees δH and ∆H , respectively. Erd˝os, Suen and Winkler [6] suggested this process as a natural probability distribution on maximal H-free graphs and asked for the typical properties of G(MH ), such as size and independence number. For the odd-cycle-free and triangle-free processes, they showed the former terminates with Θ(n2 ) edges with high probability1 , and that c1 · n3/2 ≤ MK3 ≤ c2 · n3/2 (log n) w.h.p., for positive constants c1 , c2 . Spencer [13] refined this to show the conclusion for any positive c1 , c2 , √ and conjectured that MK3 = Θ(n3/2 log n) w.h.p.. (We mention the earliest result on an H-free process is due to Ruci´ nski and Wormald [11], who showed that MK1,d+1 = bnd/2c w.h.p..) Bollob´as and Riordan [4], and, independently, Osthus and Taraz [8], considered the H-free process for the class of strictly 2-balanced H, which includes cycles, complete graphs, and complete bipartite graphs Kr,r . For such H, Bollob´as and Riordan established likely lower bounds on MH , and likely upper bounds on ∆H for H ∈ {C4 , K4 } matching the lower to within a logarithmic factor. Osthus and Taraz’s results yielded likely lower and upper bounds on MH for all strictly 2-balanced H, matching to within logarithmic factors. A further improvement came from Wolfovitz [17], who increased the lower bound on E (MH ) for regular such H. Through an application of the differential equations method (see [20] for the method and exam√ ples), Bohman [1] established Spencer’s conjecture, showing MK3 = Θ(n3/2 log n) w.h.p., and ∗

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA. Email: [email protected] 1 We say a sequence of events An occurs with high probability, or simply w.h.p., if limn→∞ Pr (An ) = 1.

1

produced an improvement on the likely lower bound for MK4 . Subsequent work by Bohman and Keevash [2] established new lower bounds on δH for all strictly 2-balanced H that hold w.h.p., and they conjectured the likely value of ∆H is at most a constant multiple of these lower bounds. For the C` -free process with ` ≥ 4, the results of Bollob´as and Riordan yielded MC` = Ω(n`/(`−1) ) and MC4 = O(n4/3 (log n)3 ) w.h.p.. Osthus and Taraz’s analysis improved the upper bound to MC` = O(n`/(`−1) · log n), and they further conjectured that the final average degree is O((n log n)1/(`−1) ). Bohman and Keevash’s results improved the lower bound by showing δC` = Ω((n log n)1/(`−1) ) w.h.p.. We mention that some interest in the H-free process comes from its connection to classical areas of extremal combinatorics. Bounds on the independence number found in [6], [13], [1], and [2] matched or improved the best known lower bounds on the Ramsey numbers R(s, t) and cyclecomplete Ramsey numbers R(C` , Kt ) for fixed s ≥ 3, ` ≥ 4 and large t, while analysis in [17] and [2] produced new lower bounds on the Tur´an numbers ex(n, Kr,r ) for fixed r ≥ 5 and large n. Further interest comes from results suggesting that the graph G(i) produced in the early stages of the H-free process resembles the random graph G(n, i), chosen uniformly at random from all graphs with n vertices and i edges (see [2] or [19]); it is also known that the final graph produced is unlikely to contain sufficiently dense subgraphs ([7] and [15]). We remark that the bounds on MH conjectured in [2] are within a constant factor of the threshold for the random graph G(n, i) to have the property that the addition of any new edge creates a new copy of H (see [12]). For graphs H containing a cycle, the problem of establishing likely bounds on MH that match to within a constant factor remains open but for a few nontrivial exceptions. Aside from cycles, these nontrivial exceptions are currently K4 , where Warnke [16] and Wolfovitz [18] independently produced upper bounds matching Bohman’s lower bound, and the diamond graph K4− [9], formed by removing an edge from K4 . Our contribution in this work is a proof that the conjecture of Bohman and Keevash (and hence of Osthus and Taraz) holds for the C` -free process for all ` ≥ 4: Theorem 1. For each ` ≥ 4, there exists a constant C = C(`) > 0 such that, with high probability, ∆C` ≤ C · (n log n)1/(`−1) . Combined with the lower bounds given by Bohman and Keevash, the following result is immediate. Corollary 1. For each ` ≥ 4, with high probability MC` = Θ(n`/(`−1) · (log n)1/(`−1) ). A second consequence of Theorem 1 is a lower bound on the independence number of G(MC` ) matching an upper bound established by Bohman and Keevash to within a constant factor. We point out that this also implies that the lower bounds for R(C` , Kt ) found in [2] will not be essentially improved by a typical graph produced by the C` -free process. Corollary 2. For each ` ≥ 4, with high probability α(G(MC` )) = Θ((n log n)(`−2)/(`−1) ). During the preparation of this paper, a proof of Theorem 1, and hence of Corollaries 1 and 2, was found independently by Warnke [14]. There are similarities between our two arguments, but an essential difference lies in which random variables we track throughout the process; this will be made more explicit below. The author [10] has also established Theorem 1 for the special case ` = 4 (independently of [14]), and the argument given here differs for the same reason.

2

1.1

Overview of the proof

For graphs H containing a cycle, the upper bounds on MH mentioned above follow from an approach used in [6] and refined in [4] and [8]: to show ∆H ≤ k, it suffices to show that in some subgraph G0 of G(M ), every set of k vertices satisfies some degree-bounding condition (DBC) which guarantees it does not lie in the neighborhood of any vertex in G(MH ). One such DBC, used in [6], [4], [8], [1], [16], [18], and [9], is that for every k-element set (or simply k-set) K and every vertex w ∈ / K, there exists a vertex v ∈ V (H) and a copy of H − v in G0 , vertex-disjoint from w, for which the neighborhood of v in H lies in K in the copy. For H = C` , this DBC is that for every k-set K and vertex w ∈ / K, there is a copy of P`−2 , the 0 path of length ` − 2, in some G ⊂ G(MC` ) which is vertex-disjoint from w and has its endpoints in K. To establish this DBC, we will produce a lower bound on the number of pairs of vertices in G(i) which can be selected as the next edge ei+1 , and which, if selected, would yield the DBC for a given k-set and vertex w. A simple union bound argument will then yield the result. This is the same approach taken by Bohman [1] to bound the size of the triangle-free process, and has subsequently been used to establish the bounds in [16], [18], [9], [10], and [14]. To bound the number of pairs of vertices which could produce the DBC for a given set K and vertex w, we will appeal to bounds on a related collection of ‘degree-bounding’ extensions (DBEs): injective mappings ϕ : {0, 1, . . . , `−2} → V (G(i))\{w} satisfying two properties. First, the vertices ϕ(0), ϕ(` − 2) lie in K. Second, there exists an index j ∗ , 0 ≤ j ∗ ≤ ` − 3, such that for all other indices 0 ≤ j ≤ ` − 3 with j 6= j ∗ , the pair ϕ(j)ϕ(j + 1) forms an edge of G(i), while the pair ϕ(j ∗ )ϕ(j ∗ + 1) is a ‘missing’ edge which can still be selected as ei+1 . In tracking such extensions, several technical difficulties arise, and we wish to mention two of them. The first is that bounds on the number of DBEs must yield suitable bounds on the pairs of vertices which would produce the DBC if selected as ei+1 . The second is that the number of DBEs for a given k-set can decrease drastically in a single step. In the triangle-free process, only the latter difficulty arises, and to handle it Bohman [1] tracked a wide collection of variables for each k-set, where each individual variable ‘ignores’ steps where sufficiently large change occurs. He obtained the desired bounds by then arguing that for each k-set, at least one of these variables experiences relatively few such ‘jumps’ overall. Analogous albeit much more sophisticated approaches were taken by Warnke [16] and by Wolfovitz [18] to handle these difficulties for the K4 -free process. Turning to the C` -free process, the author showed in [10] that these problems can be avoided in the C4 -free process by appealing to a slightly different DBC. Unfortunately, the observations allowing this to work do not easily extend to the case ` ≥ 5. In [14], Warnke shows that a careful refinement of his argument from [16] does suffice to establish Theorem 1. A common element of the preceding arguments is that each directly tracks a collection of DBEs. In this work, we show the C` -free process admits a slightly different approach. Rather than track the DBEs for each k-set, we track a collection of extensions to a path of length ` − 2 missing an edge between fixed pairs of vertices (u, v) serving as the endpoints. One barrier to ‘enough’ such extensions is a common neighbor of u and v (see Figure 1), but by ignoring pairs of vertices which have a common neighbor or which form edges, we eliminate the complications in establishing useful bounds. This approach lets us define a fairly simple collection of O(n2 ) random variables (Section 3) to which we apply the differential equations method. Then, through a simple alterations argument (similar to those used in [16] or [18]) and a few density estimates, we show these approximations

3

u

v z

Figure 1: This figure illustrates the primary difficulty caused by common neighbors, in the C6 -free process. Solid lines represent edges, and each dashed lines represents a non-edge which would complete a path of length ` − 2 = 4 between u and v if added as an edge. The presence of the common neighbor z of u and v, though, shows that each such non-edge would create a copy of C6 if added, and thus none can actually be chosen by the process. establish the DBC (Section 5). We also mention that many of the observations which simplified the analysis in [10] similarly aid our analysis when ` = 4, and we take the view that ` ≥ 5 yields the more interesting case.

1.2

Organization of the paper

The remainder of this paper is organized as follows: in Section 2, we discuss the C` -free process specifically, including results of Bohman and Keevash necessary to our argument. In Section 2.3 we introduce our main lemma, Lemma 1, and use it to prove Theorem 1. Section 3 presents the random variables we will track and the result which will establish Lemma 1, Theorem 3. Section 4 covers inequalities and lemmas necessary for our arguments, and the proof of Lemma 1 will then follow in Section 5. In Section 6 we produce additional estimates on the one-step changes in our variables necessary for the proof of Theorem 3, which follows in Section 7.

2 2.1

The C` -free process Definitions and notation

We fix an integer ` ≥ 4, and let [n] = {1, . . . , n} be the vertex set of the process, and G(i) the
graph given by the first i edges selected by the process. G(i) naturally partitions [n] 2 into three sets: E(i), O(i), and C(i). E(i) is the edge set of the process. For a pair uv ∈ / E(i), we say that uv is open, and uv ∈ O(i), if the graph G(i) + uv is C` -free; we define Q(i) := |O(i)|. Otherwise, we say uv is closed and uv ∈ C(i). For a pair of vertices uv ∈ / E(i), we define Cuv (i) to be the set of pairs xy ∈ O(i) such that G(i) + uv + xy contains a copy of C` using both uv and xy as edges. (We remark that in [2], Cuv (i) is defined analogously but consists of ordered pairs.) For each vertex v ∈ [n], we let Ni (v) and di (v) denote the neighborhood and degree, respectively, of v in G(i). For vertices u, v ∈ [n], we let disti (u, v) denote the distance between u and v in G(i), i.e., the length of a shortest path. For an integer d ≥ 0, we let Pu,v,d (i) be the number of paths of length d between u and v in G(i). We introduce a continuous time variable t, and we relate it to the process by setting t = t(i) =

4

i/n`/(`−1) . We fix constants µ, ε, W , which satisfy 0<µε

1 1  . W `

The notation 0 < a  b means there is an increasing function g(x) so the arguments which follow are valid for 0 < a < g(b). We then define the following parameters p = n−(`−2)/(`−1) , 2

s=n p=n

`/(`−1)

m = µ(log n)1/(`−1) · n`/(`−1) , ,

se = n

1/(2`)−ε

tmax = µ(log n)1/(`−1)

(1)

1/(`−1)

Γ = (n · log n)

and

,

(2)

and the following functions q(t) = e−(2t)

`−1

,

c(t) = 2(` − 1)(2t)`−2 · e−(2t)

`−1

,

`−1 +t)

f (t) = eW (t

,

(3)

noting that c(t) = −q 0 (t). All asymptotic notation, other than  above, is used under the assumption that n → ∞, and we assume that n is suitably large whenever necessary. We use the notation “±” in two distinct ways throughout this paper. For nonnegative reals a, b, the notation a ± b will denote the closed interval {a + xb : −1 ≤ x ≤ 1}. Distinct instances of ± used in interval arithmetic are treated independently, e.g., (a ± b)(c ± d) will be taken to mean {(a + x1 b)(c + x2 d) : −1 ≤ x1 , x2 ≤ 1}. We also write a = b ± c instead of a ∈ b ± c. The use of ± as a superscript, such as in Y ± (i) and y ± (t) will refer to pairs Y + (i), Y − (i) of nonnegative random variables and y + (t), y − (t) of nonnegative functions, respectively. Expressions containing such notation will represent two statements, one with all superscripted instances of ± replaced with “+”, the other with all such instances replaced with “−”. An expression such as “A± (i) = g ± (t)±γ”, combining the two uses, will denote the pair of expressions “A+ (i) = g + (t)±γ” and “A− (i) = g − (t) ± γ”.

2.2

The lower bound - results of Bohman and Keevash

Bohman and Keevash [2] established their lower bound on the C` -free process by showing that certain random extension variables are tightly concentrated throughout the initial m steps. We state their results relevant to our analysis in Theorem 2 below. To do so, in addition to bounds on the constants µ, ε, W implicit in the analysis in [2], we assume that W > (` − 1)2` ,

ε<

1 , 8(` + 1)(` − 1)

(4)

and µ < 1/2 is chosen sufficiently small that f (t) ≤ nε /2 holds for all 0 ≤ t ≤ tmax (for large n). It then follows that for all 0 ≤ t ≤ tmax , q(t)−1 ≤ q(t)−(`−3) ≤ f (t) ≤ nε /2,

and

se ≥ n7ε ,

(5)

the latter bound holding as ` ≥ 4 and (4) imply 8ε ≤ 1/(3(` + 1)) < 1/(2`). Theorem 2 (Bohman and Keevash, [2]). For each step i∗ ≥ 0, let Ti∗ denote the event that the following hold for all 0 ≤ i ≤ i∗ : 5

1.

   1 n2 f (t) − 1 q(t) ± Q(i) = 1 ± ; se se 2

(6)

∆(G(i)) ≤ Γ ;

(7)

2.

3. for all uv ∈ O(i) ∪ C(i),  |Cuv (i)| =

f (t) − 1 1± se

  4(` − 1) p−1 c(t) ± , se 2

(8)

and for all distinct uv, u0 v 0 ∈ O(i), |Cuv (i) ∩ Cu0 v0 (i)| ≤ n−1/` p−1 ; and 4. for all distinct u, v ∈ [n] and integers d, 1 ≤ d ≤ 2(` − 2), we have ( nε if d ≤ ` − 2, Pu,v,d (i) ≤ (np)d−1 pn4dε ≤ Γd−1 pn4dε if ` − 1 ≤ d ≤ 2(` − 2).

(9)

(10)

Then Tm holds with high probability. By (5), f (tmax )/se < n−6ε = o(1), 1/se = o(n−ε ), and q(tmax ) ≥ n−ε , so on the event Tm we have Q(m) > 0. This implies the process does not stop before step m, and thus Theorem 3 provides the lower bound MC` ≥ m = µn`/(`−1) (log n)1/(`−1) with high probability. We obtain (6) from Theorem 1.4 of [2] directly, and this theorem also implies the upper bound ∆(G(m)) ≤ (1 + (f (tmax ) − 1)/se )(2tmax + 1/se ) · np holds with high probability. As µ < 1/2, this bound is less than (log n)1/(`−1) · np = Γ for large n, and then (7) follows by monotonicity. Equations (8) and (9) follow from Corollary 6.2 and Lemma 8.4 of [2], respectively. We mention that the phrasing of Lemma 8.4 suggests that uv and u0 v 0 are fixed. However, as Lemma 8.4 is shown to be a consequence of a constant (depending on H) number of applications of Lemma 5.2, which has exponentially small failure probability, (9) follows for all such pairs and steps i, 0 ≤ i ≤ m, with high probability. Finally, as (4) implies ε < (2·(2(`−2))·`)−1 , (10) follows from Lemma 5.1 of [2]. In the notation of [2], for the path of length d, Pd , with vertex set {0, 1, . . . , d} and edge set {01, 12, . . . , (d−1)d}, by letting A = {0, d} and J = Pd , A is independent in Pd and the pair (A, J) is ‘strictly balanced’ with respect to the C` -free process. For fixed u, v ∈ [n], letting φ : {0, d} → {u, v} via φ(0) = u, φ(d) = v, the quantity Nφ,J is precisely Pu,v,d (m), and the parameter SA,J = (np)d−1 p is less than 1 if and only if d < ` − 1. The proof of Lemma 5.1 yields an exponentially small failure probability, which more than suffices to cover our O(n2 ) applications. Further improvements of these bounds, such as noting that for d ≥ `, Pu,v,d (i) is ‘trackable’ in the sense of [2], are possible but unnecessary.

2.3

The upper bound and independence number

We begin by fixing a constant C = C(`) and size k such that   4 · (` − 3)! , and k = C · (n log n)1/(`−1) = C · Γ. C > max 56, µ`−2 6

(11)

Recalling the DBC for the C` -free process, for a step i, a set of vertices A ⊆ [n] and a vertex w ∈ [n] \ A, we say that A is w-covered in G(i) if there exists a copy of P`−2 in G(i) which is vertex-disjoint from w and has its endpoints in A. We say A is w-uncovered in G(i) otherwise.
To prove Theorem 1, we will show that every K ∈ [n] k is w-covered in G(m) for all w ∈ [n] \ K w.h.p..
For each step 0 ≤ i ≤ m, set K ∈ [n] k and vertex w ∈ [n] \ K, let RK,w (i) ⊆ O(i) be the set of open pairs xy such that G(i) + xy contains a copy of P`−2 which excludes w, includes the edge xy, and has its endpoints in K. These are the open pairs whose selection as ei+1 would result in K being w-covered for all steps i0 ≥ i + 1. Our main lemma establishes a lower bound on |RK,w (i)| in the range m/2 ≤ i ≤ m. Lemma 1. For i∗ ≥ m/2, let Ri∗ denote the event that for all steps i, m/2 ≤ i ≤ i∗ , all sets of
vertices K ∈ [n] k , and all vertices w ∈ [n] \ K, if K is w-uncovered in G(i) then |RK,w (i)| ≥

(2tnp)`−3 q(t)k 2 . 2 · (` − 3)!

(12)

Then Rm holds with high probability. The proof of Lemma 1 is given in Section 5; we use it now to prove Theorem 1.
Proof of Theorem 1. We begin by fixing a step m/2 ≤ i ≤ m − 1, a set of vertices K ∈ [n] k , and a vertex w ∈ [n] \ K. Suppose that Ti and Ri hold, and that K is w-uncovered in G(i). As ei+1 is chosen uniformly at random from Q(i) open pairs, the probability that K is w-uncovered in G(i+1) |R (i)| is 1 − K,w Q(i) , which bounds above the probability that Ti+1 and Ri+1 additionally hold. Applying (12) and bounding Q(i) ≤ q(t) · n2 (valid by (5) and (6) for large n), we have |RK,w (i)| (2tnp)`−3 q(t) · k 2 µ`−3 C 2 Γ`−1 µ`−3 C 2 log n ≥ ≥ · 2 = · 2 Q(i) 2(` − 3)! · q(t)n 2(` − 3)! n 2(` − 3)! n since k = C · Γ, Γ`−1 = n log n, and, as i ≥ m/2, 2tnp ≥ tmax np = µ · Γ. Consequently, the probability that Ti and Ri hold and K remains w-uncovered for all steps m/2 ≤ i ≤ m is at most 

µ`−3 C 2 log n 1− · 2(` − 3)! n

m/2

    µ`−3 C 2 log n m µ`−2 C 2 ≤ exp − · · = exp − · Γ log n . 2(` − 3)! n 2 4(` − 3)!

Since Ti and Ri are decreasing, a union bound over at most nk+1 = exp((1 + o(1)) · C · Γ log n) choices of K and w gives that the probability Tm and Rm hold and a w-uncovered k-set exists in G(m) is o(1) by (11). Since the probability that Tm or Rm fails is o(1) by Theorem 3 and Lemma 1, we conclude that every k-set K is w-covered in G(m) for every vertex w ∈ [n] \ K with high probability, and thus ∆C` ≤ k w.h.p.. Corollary 1 is immediate, so we next prove Corollary 2. Proof of Corollary 2. Let M = MC` . The upper bound on α(G(M )) follows from Theorem 1.9 of [2]. For the lower bound, we appeal to the argument given in [10] for the case ` = 4, which requires a bound on the number of triangles in G(M ). Since G(M ) is C` -free, for each vertex v ∈ [n], the 7

subgraph induced by NM (v) is P`−2 -free, and so has at most `−3 2 |NM (v)| < `∆C` edges by a result of Erd˝os and Gallai [5]. Thus, each vertex lies on at most `∆C` triangles, and so G(M ) contains less than `∆C` · n triangles in total. By Theorem 1, ∆C` < k with high probability, so by applying Lemma 12.16.(ii) of [3] with d = k and h = `kn yields that for n sufficiently large,      1 n n log k 1 `kn α(G(M )) ≥ ≥ · · log k − log = Ω (n log n)(`−2)/(`−1) . 10 k 2 n 30 · k

3

Random variables

We next present the random variables we track in order to establish Lemma 1. We fix a step i, 0 ≤ i ≤ m, and we suppose the event Ti of Theorem 2 holds. We begin with some heuristic motivation, supposing the graphs G(i) produced by the C` -free process resemble the binomial
random graph G(n, 2tp), in which edges are chosen independently with probability 2tp ≈ i/ n2 . We note that q(t) is the asymptotic probability in G(n, 2tp) that a given pair vertices is ‘open’ i.e., that it neither forms an edge nor would create a copy of C` if added as an edge. Fixing an ordered pair (u, v) of vertices, our goal is to bound the number of open pairs whose selection as ei+1 would complete a path of length ` − 2 between u and v. We do so by tracking a collection of ‘incomplete’ paths, missing an edge, of length ` − 2 between u and v, such that the missing edge is incident with v and is an open pair. Each such incomplete path consists of (complete) path of length ` − 3 in G(i) from u to some vertex z 6= v, where the pair zv is open. Consequently, the (heuristic) expected number of such incomplete paths is ≈ n`−3 (2tp)`−3 · q(t). We next note that the expected number of paths of length at most ` − 3 between two vertices u and z is (heuristically) at most P P`−3 a−1 −1+a/(`−1)+o(1) = o(1). (2tp)a = `−3 a=1 n a=1 n This suggests that if a path of length at most ` − 3 exists between two vertices, it is likely to be a unique shortest-distance path. By imposing this as a requirement for the incomplete paths we track, each such path will necessarily count a distinct open pair, at an asymptotically negligible cost to the (heuristic) calculations above. If we further restrict ourselves to consider those incomplete paths that are constructed edge-by-edge starting at u as the process evolves, we greatly reduce the number of intermediate variables that we must track at a minor cost of reducing our estimate by a factor of (` − 3)!. These heuristic arguments are likely to fail if u and v share a common neighbor z: in this case, for most such incomplete paths, the missing edge would complete a copy of C` (using u, v, z as vertices) if added, and therefore must be closed. (See Figure 1.) However, provided u and v are distance at least 3 apart in G(i), the claimed estimate will be shown to hold. Turning to the formal definition of our variables, for each step i ≥ 0 we define the set     [n] U (i) = xy ∈ : disti (x, y) ≤ 2 , 2 recalling disti (x, y) denotes the distance between x and y in G(i). We also define, for each edge xy ∈ E(i), ind(xy) to be the integer r, 1 ≤ r ≤ i, such that xy = er . 8

Next, for each step i ≥ 0, ordered pair of distinct vertices (u, v) satisfying uv ∈ / U (i), and integer j, 0 ≤ j ≤ ` − 3, we define Xu,v,j (i) to be set of all injective maps ϕ : {0, 1, . . . , ` − 2} → V (G(i)) = [n], where, letting ϕa denote ϕ(a), we have ϕ0 = u and ϕ`−2 = v, and the following five additional conditions: (i) {ϕ0 ϕ1 , ϕ1 ϕ2 , . . . , ϕj−1 ϕj } ⊂ E(i); (ii) {ϕj ϕj+1 , ϕj+1 ϕj+2 , . . . , ϕ`−3 ϕ`−2 } ⊂ O(i); (iii) ind(ϕ0 ϕ1 ) < ind(ϕ1 ϕ2 ) < · · · < ind(ϕj−1 ϕj ); (iv) for 0 ≤ a ≤ j, (u = ϕ0 , ϕ1 , ϕ2 , . . . , ϕa ) is the unique path of length at most a between u and ϕa in G(i); and (v) for j + 1 ≤ a ≤ ` − 3, disti (u, ϕa ) > a. We extend this definition to all ordered pairs of vertices (u, v) by setting Xu,v,j (i) = Xu,v,j (i − 1) if uv ∈ U (i). Figure 2 provides an illustration of this definition. We will refer to the maps in Xu,v,j (i) as extensions, noting the extensions in Xu,v,`−3 (i) are the incomplete paths we set out to count. The role of each set Xu,v,j (i), 0 ≤ j ≤ ` − 4 is to allow us to track the one-step changes to the set Xu,v,j+1 (i): (i) and (ii) ensure that elements of Xu,v,j (i) can extend to the desired path, while (iii) fixes the order in which the path is built. Condition (iv)’s primary purpose is to ensure that each extension ϕ ∈ Xu,v,`−3 (i) counts a distinct open pair ϕ`−3 v, as desired. The role of (v) is two-fold. First, it guarantees that for j ≥ 1, if ϕ ∈ Xu,v,j−1 (i) and ei+1 = ϕj ϕj+1 is the ‘next’ edge in the incomplete path, then (u = ϕ0 , ϕ1 , . . . , ϕj , ϕj+1 ) is a unique shortestdistance path in G(i+1), so ϕ satisfies (iv) for the definition of Xu,v,j (i+1). In conjunction with (i)(iv), (v) also guarantees that any extension θ ∈ Xu,v,j (i + 1) \ Xu,v,j (i) must satisfy θ ∈ Xu,v,j−1 (i). (This will be argued in the proof of Lemma 12 in Section 6.2.) Thus, this collection of sets is ‘sufficient’ to track the set Xu,v,`−3 (i) as the process evolves. As mentioned in the introduction, this collection of random variables is similar to that analyzed by Warnke in [14]. In addition to (i) and (ii), the variables considered in [14] also satisfy (iii): our use of this condition was inspired by a similar use in [16]. We mention that condition (iv) is similar to a condition imposed in [16] to ensure that the tracked extension variables count distinct open pairs, and a variant of that property is also employed in [14]. A similar heuristic argument as given above suggests that for 0 ≤ j ≤ ` − 3, we should expect |Xu,v,j (i)| ≈

(2t)j q(t)`−2−j 1 · n`−3 (2tp)j q(t)`−2−j = · (np)j n`−3−j . j! j!

Defining Sj = Sj (n) := (np)j n`−3−j

and

xj (t) :=

(2t)j q(t)`−2−j , j!

our main tool for establishing Lemma 1 is that this is the ‘right’ approximation.

9

u = ϕ0

ϕ1

ϕ2

ϕj−1

ϕj

ϕj+1

ϕj+2

ϕ`−3

ϕ`−2 = v

Figure 2: Our view of the typical extension ϕ ∈ Xu,v,j (i); solid lines represent edges of G(i) and dashed lines represent open pairs. The reader may find it helpful to view ϕ as consisting of a (unique, shortest-distance) path of length j starting at u, along with a collection of ` − 3 − j ‘free’ vertices ϕj+1 , . . . , ϕ`−3 . This simplified viewpoint will aid in later calculations. Theorem 3. With high probability, for all steps 0 ≤ i ≤ m, indices 0 ≤ j ≤ ` − 3, and ordered pairs (u, v) of distinct vertices in [n] such that uv ∈ / U (i),   f (t)q(t)−j |Xu,v,j (i)| = xj (t) ± Sj . (13) n3ε The proof of Theorem 3, an application of the differential equations method, is given in Section 7.

4

Preliminaries

Our aim in this section is to present various bounds to be used in our later arguments. We begin by noting the following simple relationships between Γ and n, p, and Sj for 0 ≤ j ≤ ` − 3: n=

Γ`−1 , log n

Γj n`−3−j ≤ (log n) · Sj ,

and

p = n−(`−2)/(`−1) ≤

log n . Γ`−2

(14)

Since Γ → ∞ as n → ∞, for any fixed positive integer j we may also assume 1 + Γ + · · · + Γj ≤ 2Γj . To facilitate certain calculations in Section 7.2, we note that as tmax = µ · (log n)1/(`−1) , for any fixed polynomial F (t) of degree less than `−1, |F (t)| = o(log n) uniformly on [0, tmax ]. Consequently, c(t) ≤

c(t) = o(log n) q(t)

xj (t) ≤

and

xj (t) ≤ (2t)j = o(log n) q(t)

for 0 ≤ j ≤ `−3 and t ∈ [0, tmax ]. More generally, if F (t) is any fixed polynomial, then |F (t)| = o(nε ) uniformly on [0, tmax ]. We also note the following simple lemma taken from [10]. Lemma 2. Suppose η = η(n) → 0 as n → ∞, and a, b are positive integers. Then, for n sufficiently large, 1. (1 ± aη)(1 ± bη) ⊆ (1 ± (a + b + 1)η). 2. For all x ∈ (1 ± aη) , x−1 ∈ (1 ± (a + 1)η). Proof. Both follow from η 2 = o(η), the latter from considering the series expansion of (1+z)−1 .

4.1

The number of extensions containing given vertices

Our aim in this section is to present Lemma 3 below, which bounds the number of extensions lying in the set Xu,v,j (i) which include a given vertex or which ‘count’ a given open pair. To do so, we introduce some additional notation. 10

First, for a vertex w ∈ [n] and an integer 0 ≤ a ≤ ` − 2, let Φiu,v,j (w; a) ⊆ Xu,v,j (i) be the set of extensions in Xu,v,j (i) which map a to the vertex w, so Φiu,v,j (w; a) = {ϕ ∈ Xu,v,j (i) : ϕa = w}. Since every extension in Xu,v,j (i) is an injective map satisfying ϕ0 = u and ϕ`−2 = v, we have, for example, Φiu,v,j (u; 0) = Xu,v,j (i) = Φiu,v,j (v; ` − 2), and Φiu,v,j (w; a) = ∅ for all other choices of (w, a) ∈ {u, v} × {0, 1, . . . , ` − 2}. We next let Φiu,v,j (w) ⊆ Xu,v,j (i) be the collection of all extensions in Xu,v,j (i) which include w in their image, so `−2 [ Φiu,v,j (w) = Φiu,v,j (w; a). (15) a=0

Φiu,v,j (u)

Φiu,v,j (v).

Thus, for example, = Xu,v,j (i) = Finally, for an open pair xy ∈ O(i), we define the set Φiu,v,j (xy) = {ϕ ∈ Xu,v,j (i) : xy = ϕa ϕa+1 for some a, j ≤ a ≤ ` − 3}. These are the extensions ϕ ∈ Xu,v,j (i) which include the open pair xy in the collection of open pairs {ϕj ϕj+1 , ϕj+1 ϕj+2 , . . . , ϕ`−3 ϕ`−2 } ‘counted’ by ϕ. We mention that bounds on the quantity |Φiu,v,j−1 (ei+1 )| will imply bounds on the maximum one-step increase in the variables |Xu,v,j (i)|. To bound the size of these sets in Lemma 3, we take a simplified view of the extensions under consideration. Specifically, we view a given extension ϕ ∈ Xu,v,j (i) as consisting of a path of length j in G(i) starting at u, (ϕ0 = u, ϕ1 , . . . , ϕj ), along with an ordered (` − 3 − j)-tuple of ‘free’ vertices (ϕj+1 , . . . , ϕ`−3 ). (See Figure 2.) For example, we may bound the total number of extensions in Xu,v,j (i) by first bounding the number of ways to construct a path of length j starting at u, and then select the ‘free’ vertices in at most n ways each, yielding |Xu,v,j (i)| ≤ ∆(G(i))j · n`−3−j . Lemma 3. Provided n is sufficiently large, for all steps 0 ≤ i ≤ m, if ∆(G(i)) ≤ Γ then the
following holds. For all pairs uv ∈ [n] 2 \ U (i) and all vertices w ∈ [n] \ {u, v}, if 0 ≤ j ≤ ` − 3 and 1 ≤ a ≤ ` − 3, then ( Γj−a n`−3−j ≤ ((log n) · Γ−a ) · Sj if 1 ≤ a ≤ j, i |Φu,v,j (w; a)| ≤ (16) Γj n`−3−j−1 ≤ ((log n) · n−1 ) · Sj if j + 1 ≤ a ≤ ` − 3. Consequently, |Φiu,v,j (w)| ≤

(` − 3) log n · Sj . Γ

(17)

Furthermore, for each open pair xy ∈ O(i), we have |Φiu,v,`−3 (xy)| ≤ 1, and if 0 ≤ j ≤ ` − 4, |Φiu,v,j (xy)|

 ≤

2` log n n

 · Sj .

(18)

Proof. We begin by fixing distinct vertices u, v, w ∈ [n] and indices 0 ≤ j ≤ ` − 3 and 1 ≤ a ≤ ` − 3. Suppose first that j + 1 ≤ a ≤ ` − 3: we may bound the number of paths of length j starting at u above by ∆(G(i))j (selecting ϕ1 , . . . , ϕj in order so that ϕi−1 ϕi ∈ E(i)), and then the number of selections of the ` − 3 − j − 1 ‘free’ vertices in {ϕj+1 , . . . , ϕ`−3 } \ {ϕa } in at most n`−3−j−1 ways. 11

This yields |Φiu,v,j (w; a)| ≤ ∆(G(i))j · n`−3−j−1 , and the corresponding bounds in (16) follow from (7) and (14). Suppose instead that 1 ≤ a ≤ j, and fix an extension θ ∈ Φiu,v,j (w; a). (If none exists, (16) is trivial.) By (iv), the path (u = θ0 , θ1 , . . . , θa = w) is the unique path of length a between u and w in G(i), and so every extension ϕ ∈ Φiu,v,j (w; a) satisfies (ϕ0 , . . . , ϕa ) = (θ0 , . . . , θa ). We may therefore bound the total number of such extensions ϕ by noting there are less than ∆(G(i))j−a choices for the remainder of the path (ϕa+1 , . . . , ϕj ) and at most n`−3−j choices for the ‘free’ vertices, so |Φiu,v,j (w; a)| ≤ ∆(G(i))j−a · n`−3−j . The remainder of (16) then follows from (7) and (14) as above. We obtain (17) by directly applying the bounds in (16) to (15), noting that |Φiu,v,0 (w)| = 0 = |Φiu,v,`−2 (w)| as w ∈ / {u, v}. To prove (18), suppose xy ∈ O(i). Supposing j = ` − 3, any extension ϕ ∈ Φiu,v,`−3 (xy) must satisfy xy = ϕ`−3 ϕ`−2 = ϕ`−3 v. Without loss of generality we may assume y = v = ϕ`−2 , implying x = ϕ`−3 , and thus |Φiu,v,`−3 (xy)| = |Φiu,v,`−3 (x; ` − 3)| ≤ 1 by (16) (or by (iv) directly). If, instead, 0 ≤ j ≤ `−4, then as each extension ϕ ∈ Φiu,v,j (xy) satisfies xy ∈ {ϕj ϕj+1 , . . . , ϕ`−3 ϕ`−2 }, at least one of x or y lies in the set {ϕj+1 , . . . , ϕ`−3 } of ‘free’ vertices. It follows that |Φiu,v,j (xy)| ≤

`−3 X


|Φiu,v,j (x; a)| + |Φiu,v,j (y; a)| ,

a=j+1

and then (18) is implied by (16).

4.2

Three technical lemmas

In this subsection we present three technical lemmas for our later arguments. The first of these will be used in our proof of Lemma 1 in Section 5.2 to show that few open pairs are ‘overcounted’ when we translate the estimates on the variables |Xu,v,`−3 (i)| given in Theorem 3 into lower bounds on |RK,w (i)|. Our statement only considers ` ≥ 5, as the situation it resolves does not arise in the C4 -free process. Lemma 4. Suppose ` ≥ 5, Ti holds, and let u, x ∈ [n] be distinct vertices. Let Z be the set of vertices z ∈ [n] such that disti (u, z) = disti (x, z) = ` − 3, and such that for all paths (u = u0 , u1 , u2 , . . . , u`−3 = z) of length ` − 3 in G(i), xu1 ∈ / E(i). For n sufficiently large, |Z| ≤ 8(`−1)ε `−5 n ·Γ . Proof. We begin by fixing a vertex z ∈ Z: let (u = u0 , . . . , u`−3 = z) and (x = x0 , x1 , . . . , x`−3 = z) be shortest-distance paths in G(i). Let b be the least index such that xb ∈ {u0 , u1 , . . . , u`−3 }, and let a be the index so that ua = xb . Then (u0 , u1 , . . . , ua = xb , xb−1 , . . . , x0 ) is a path of length a + b in G(i) between u and x, and as ` − 3 − a = disti (ua , z) = disti (xb , z) = ` − 3 − b, it follows that b = a. Furthermore, as u 6= x and as, by assumption, u1 x ∈ / E(i), we also have a ≥ 2. It follows that each vertex z ∈ Z corresponding to a given value of a, 2 ≤ a ≤ ` − 3, as found above must lie at the end of a path of length ` − 3 − a from the central vertex on a path of length P`−3 `−3−a . So, applying (10), we have 2a between u and x. As Ti holds, |Z| ≤ a=2 Pu,x,2a (i) · Γ `−3−a ε `−5 Pu,x,2a (i) · Γ ≤n ·Γ if 2a ≤ ` − 2 and Pu,x,2a (i) · Γ`−3−a ≤ (Γa+1 pn8aε )Γ`−5 ≤ (Γ`−2 pn8(`−3)ε )Γ`−5 ≤ (n8(`−3)ε log n)Γ`−5 12

if ` − 1 ≤ 2a ≤ 2(` − 3), the last bound following from (14). Combining, we see   |Z| ≤ ` nε + n8(`−3)ε log n Γj−2 ≤ n8(`−1)ε Γ`−5 for n sufficiently large. The second technical lemma, Lemma 5 is similar to Lemma 4, and will be applied in Section 6 in our estimates of the expected one-step increase and the maximum one-step decrease in the variables |Xu,v,`−3 (i)|. Lemma 5. Suppose that Ti holds, and let j and d be integers satisfying ` − 4 ≤ j ≤ d ≤ ` − 2. Let u, x ∈ [n] satisfy disti (u, x) > d − j, and let Z be the set of vertices z ∈ [n] such that disti (u, z) = j and a path of length d between x and z exists in G(i). For n sufficiently large, |Z| ≤ n8(`−1)ε · Γj−1 . Proof. Let z ∈ Z be arbitrary. Let (u = u0 , u1 , . . . , uj = z) be a shortest path in G(i) between u and z, and let (x = x0 , x1 , . . . , xd = z) be any path of length d between x and z in G(i). Let b be the least index such that xb ∈ {u0 , u1 , . . . , uj }, so b ≤ d, and let a ≤ j be the index such that ua = xb . Minimality of b implies (u0 , u1 , . . . , ua = xb , xb−1 , . . . , x0 ) is a path in G(i) of length a + b, and so a + b ≥ disti (u, x) > d − j. On the other hand, (ua , . . . , uj = z) is a shortest path in G(i), so as ua = xb and z = xd , it follows that d − b ≥ j − a, or, equivalently, b ≤ a + (d − j). Let A = {(a, b) : a + b ≥ disti (u, x), b ≤ a + (d − j), a ≤ j, b ≤ d} be the pairs of nonnegative integers satisfying the prior inequalities. We first bound the number of vertices z ∈ Z which produce a given pair (a, b) ∈ A through the construction above. By that construction, we see each such vertex z lies at distance j − a from a given vertex ya on a path (u = y0 , . . . , ya , . . . , ya+b = x) of length a + b between u and x in G(i). Recalling Pu,x,a+b (i) is the number of paths of length a + b between u and x in G(i), as Ti holds, at most Pu,x,a+b (i) · Γj−a vertices z ∈ Z produce (a, b). For each (a, b) ∈ A, as a + b ≥ disti (u, x) > d − j and b ≤ a + (d − j), it follows that a > 0, and we also note that a + b ≤ 2(` − 2) as a ≤ j ≤ ` − 2 and b ≤ d ≤ ` − 2 by assumption. Now, if a + b ≤ ` − 2, then by (10) and a > 0, Pu,x,a+b · Γj−a ≤ nε · Γj−1 . If a + b ≥ ` − 1, then as b ≤ ` − 2, we again apply (10) to obtain Pu,x,a+b (i) · Γj−a ≤ (Γa+b−1 pn4(a+b)ε )Γj−a ≤ (Γ`−2 pn8(`−2)ε )Γj−1 . Since |A| ≤ `2 and since Γ`−2 p ≤ log n by (14), it follows that for n suitably large,   X |Z| ≤ Pu,x,a+b (i) · Γj−a ≤ `2 nε + (log n)n8(`−2)ε · Γj−1 ≤ n8(`−1)ε · Γj−1 . (a,b)∈A

We end this section with a simple estimate on the number of open pairs which can complete a ‘short’ path between two given vertices in G(i). These bounds will be used to show that the contribution to the expected one-step decrease in the variables |Xu,v,j (i)| due to conditions (iv) or (v) failing is negligible in comparison to the contribution due to condition (ii) failing. Lemma 6. Suppose ∆(G(i)) ≤ Γ, and let x, y ∈ [n] be distinct vertices, and let j be an integer, 1 ≤ j ≤ ` − 3. Then the number of open pairs z0 z1 ∈ O(i) which lie on a path of length exactly j between x and y in G(i) + z0 z1 is at most jΓj−1 , and, provided n is suitably large, the number which lie on a path of length at most j is at most 2`Γj−1 . 13

Proof. For a pair z0 z1 to lie on a path of length exactly j between x and y, G(i) must contain a path of length d from x to, say, z0 , and a path of length ` − d − 1 from z1 to y. Thus, for each d there are at most Γd+(j−d−1) ways to select such a pair z0 z1 , and hence at most jΓj−1 ways total, yielding the first bound. Summing this from 1 to j yields the second bound, as j ≤ `.

5

Proof of Lemma 1

Our aim in this section is to prove Lemma 1, and we begin with heuristic motivation for our argument. Fix a step i, m/2 ≤ i ≤ m, a k-element set K ⊆ [n] of vertices, recalling k = C · Γ for a suitably large constant C, and a vertex w ∈ [n] \ K. Suppose K is w-uncovered in G(i), so there does not exist a path of length ` − 2 with endpoints in K and which is vertex-disjoint from w. Finally, recall that RK,w (i) is the set of open pairs xy ∈ O(i) whose selection as ei+1 produces such a path in G(i + 1). Let     K K = (u, v) : uv ∈ \ U (i) , 2 and let (u, v) ∈ K. Each extension ϕ ∈ Xu,v,`−3 (i) counts an open pair ϕ`−3 ϕ`−2 = ϕ`−3 v which, if selected as ei+1 , would complete a path of length ` − 2 between u and v. Recalling Φiu,v,`−3 (i) ⊆ w := X Xu,v,`−3 (i) consists of those extensions including w in their image, if ϕ ∈ Xu,v u,v,`−3 (i) \ i Φu,v,`−3 (w), then the counted open pair ϕ`−3 v lies in RK,w (i). That is, [ [ RK,w (i) ⊇ {ϕ`−3 ϕ`−2 }. (u,v)∈K

w ϕ∈Xu,v

We note the lower bound (12) we aim to prove is |RK,w (i)| ≥ (k 2 /2) · x`−3 (t)S`−3 , and Theorem 3 w | = (1 − o(1))) · x and Lemma 3 imply |Xu,v `−3 (t)S`−3 for all (u, v) ∈ K w.h.p.. Thus, the number of S w extensions | (u,v)∈K Xu,v | considered in the collection above is ≈ |K| · x`−3 (t)S`−3 . Furthermore, simple density estimates imply |K| > k 2 /2 with room to spare. So, to show (12) follows, we will argue that o(|K| · x`−3 (t)S`−3 ) pairs of extensions considered above count the same open pair, implying the majority of such extensions count unique open pairs. One potential source of overcounting comes from extensions which count an open pair conw without affecting the tained in K, but Lemma 3 implies these can be excluded from the sets Xu,v asymptotics above. After removing such extensions, suppose ϕ and θ are two remaining, distinct extensions which count the same open pair ϕ`−3 ϕ`−2 = θ`−3 θ`−2 . It follows that ϕ`−3 = θ`−3 = z w and for some vertex z ∈ [n] \ K, and θ`−2 = ϕ`−2 = v for some vertex v ∈ K. Thus, ϕ ∈ Xu,v θ ∈ Xuw0 ,v for some vertices u, u0 ∈ K. Condition (iv) implies u 6= u0 as ϕ 6= θ, and further guarantees that once the shortest paths from u to z and from u0 to z intersect, they coincide. That is, for some integer a, 1 ≤ a ≤ ` − 3, ϕb = θb holds for a ≤ b ≤ ` − 3, and {ϕ0 , . . . , ϕa−1 } ∩ {θ0 , . . . , θa−1 } = ∅. Thus, z lies at the end of a path of length ` − 3 − a from the central vertex of a path of length 2a between u and u0 , and so we can bound the number of pairs of extensions (ϕ, θ) which count the same open pair by bounding the number of choices of such vertices u, u0 , v, z. Assuming G(i) behaves like G(n, 2tp), in expectation this is at most 3

k ·

`−3 X

n2a−1 (2tp)2a · n`−3−a (2tp)`−3−a ≤ k 3 · (` − 3)(2t)2`−6 (np)`−4 S`−3 · p

a=1

14

  = k 2 · n(`−3)/(`−1)+o(1) · p · S`−3 . By (1), n(`−3)/(`−2)+o(1) · p = n−1/(`−1)+o(1) . However, (5) and i ≥ m/2 imply x`−3 (t) = Ω(n−ε ), so as ε < 1/(` − 1) by (4), the number of pairs of extensions counting the same open pair is (heuristically) negligible compared with the total number of extensions in the collection, as desired. We can nearly make this argument fully rigorous using the estimates provided in Theorem 2 in place of expected value calculations. However, in the case a = 1, these bounds are too large to establish the result, and additional care is necessary. We handle this by passing to a carefully chosen subset K 0 ⊆ K and corresponding (large) subset K0 ⊆ K. We then show that a small w with (u, v) ∈ K0 so that if two extensions number of extensions can be removed from each set Xu,v w and θ ∈ X w remain and count the same open pair, the path between u and u0 (given by ϕ ∈ Xu,v u0 ,v θ and ϕ) has length at least 4. The bounding argument given above (presented as an application of Lemma 4) then suffices. The construction of this subset K 0 is the subject of our next technical lemma, Lemma 10, presented in the next subsection.

5.1

Preparations for the proof

Our aim in this section is to present a technical lemma, Lemma 10 below, which shows that each 0
suitably large w-uncovered set A contains a large subset A0 for which | A2 ∩ U (i)| is relatively small, and such that for each vertex v ∈ A0 , few of v’s neighbors in G(i) have any other neighbors in A0 . This latter condition will allow us to handle the overcounting issue discussed above. We begin with an estimate on the probability of subgraph containment in G(m) from [2], and mention that the proof given in [2] holds in our analysis as our event Tm implies the lower bound Q(i) ≥ n2−ε /2 for all 0 ≤ i ≤ m. (This follows for large n from (5) and (6).) Lemma 7 ([2], Lemma 4.1). For any fixed graph F on [n] with eF edges, the probability that Tm holds and F ⊆ G(m) is at most (pn2ε )eF . Next, we establish bounds on the number of vertices in G(i) which have large degree in a given set A. Following [2], for a set A ⊆ [n] and integer d ≥ 0, we define the sets of vertices DA,d = {v ∈ [n] : |Nm (v) ∩ A| ≥ d}

and

0 = DA,d \ A. DA,d

Our next lemma, Lemma 4.3 of [16], follows from Lemma 7 with the same proof as in [16]. Lemma 8 ([16] Lemma 4.3). For any d ≥ max{16ε−1 , 2apn2ε }, the probability that Tm holds and there exists an A ⊆ [n] with |A| = a and |DA,d | ≥ 16ε−1 d−1 a is at most n−a . We will only require the following corollary of Lemma 8, noting that 2(Γ log n)pn2ε = o(1) by the final bound in (14) since ` ≥ 4 and as 2ε < 1/(` − 1) by (4). Corollary 3. The probability that Tm holds and there exists an A ⊆ [n], Γ ≤ |A| ≤ Γ log n, with |DA,log n | ≥ 16ε−1 |A|/(log n) is at most (Γ log n) · n−Γ . We will also utilize a variant of Lemma 8 to bound the number of pairs of vertices in a wuncovered set A with a common neighbor outside A if ` ≥ 5. (We obtain a suitable bound when ` = 4 directly from the definition of a w-uncovered set.) The proof of this next result is based on the proof of Lemma 4.2 of [2]. 15

Lemma 9. Let ` ≥ 5. The probability that Tm holds and there exists a set A ⊆ [n], Γ ≤ |A| ≤ 0 | ≥ 13|A|/d is at most Γ2 (log n) · n−Γ/(4`−4) . Γ log n, and an integer d, 2 ≤ d ≤ Γ, for which |DA,d Proof. We fix integers a and d satisfying Γ ≤ a ≤ Γ log n and 2 ≤ d ≤ Γ, and we let b = d13a/de. 4`−7 We note that 13 ≥ 2`−9 as ` ≥ 5. By the union bound, it suffices to show that on the event Tm , 0 | ≥ b occurs for some A ⊆ [n] with |A| = a is at most n−Γ/(4`−4) . the probability pa,d that |DA,d By the union bound and Lemma 7, pa,d

b b    d+1    b   n n n a n e n ne(ae)d (pn2ε )d 2ε d 2ε db · · (apn ) . ≤ ≤ (pn ) ≤ a b d a a bdd 13dd−1 a

Noting that d ≥ 2 implies ed+1 /dd−1 ≤ e3 /2 < 13, that 3ε < 1/(4(` − 1)) and that a ≥ Γ ≥ e · np for large n, the bound follows as
log pa,d ≤ a log(ne/a) + b log(n/a) + d log(apn2ε )   `−2 1 `−2 ≤ a log(ne/a) + bd + − + 3ε log(n) d(` − 1) ` − 1 ` − 1   2` − 9 ≤ a log(ne/a) + bd − log(n) 4(` − 1)   Γ ` − 2 13(2` − 9) − log(n) ≤ − · log n. ≤a `−1 4(` − 1) 4` − 4 With the pieces in place, we turn to the desired lemma, which we state for all w-uncovered sets
A in a range of sizes including k. Properties (a) and (b) imply suitable bounds on | A2 \ U (i)|, while (c) relates to the issue of overcounting open pairs. Lemma 10. For all steps i∗ ≥ 0, let Di∗ be the event that the following holds for all steps 0 ≤ i ≤ i∗ . For every set A ⊆ [n] and every vertex w ∈ [n]\A such that 2Γ ≤ |A| ≤ log7 n ·Γ and A is w-uncovered in G(i), there exists a subset A0 ⊆ A satisfying the following three properties: (a) |A0 | ≥ (1 − 120ε−1 /(log n)) · |A|, 0
(b) | A2 ∩ U (i)| ≤ 14Γ|A0 |, and (c) for each v ∈ A0 , at most 2 log n vertices z ∈ Ni (v) satisfy z ∈ A0 or |Ni (z) ∩ A0 | ≥ 2. Then the probability that Tm holds and Dm fails is at most n−Γ/(4`) = o(1). Proof. We consider the case ` = 4 first, and assume n is sufficiently large throughout. We fix a step 0 ≤ i ≤ m, a set A ⊆ [n] with 2Γ ≤ |A| ≤ log7 n Γ, and a vertex w ∈ [n] \ A. In this case, we will show that on the event Ti , if A is w-uncovered in G(i) then (a)-(c) hold for A0 = A, and so Tm implies Dm deterministically. In this case, (a) is immediate, and if A is w-uncovered then |Ni (u)∩A| ≤ 1 for all u ∈ [n]\{w} by definition. So, the subgraph of G(i) induced by A has maximum degree at most 1, and w is the only vertex which could have at least two neighbors in A, implying (c). As A has at most |A|/2 < Γ|A| edges and as w is the only possible common neighbor of at most |Ni (w) ∩ A|2 < Γ2 < Γ|A| pairs of
vertices in A, |U (i) ∩ A2 | ≤ 2Γ|A| follows and (b) holds. 16

Now, we suppose ` ≥ 5 and assume Tm holds. We also assume that |DB,log n | ≤ 16ε−1 |B|/(log n) 0 | ≤ 13d−1 |B| for all B ⊆ [n] with Γ ≤ |B| ≤ (log n) · Γ and 2 ≤ d ≤ Γ. This is valid and |DB,d as Corollary 3 and Lemma 9 yield the collective failure probability on Tm of these bounds is less than n−Γ/(4`) . We will show that each set A ⊆ [n] with 2Γ ≤ |A| ≤ log7 n · Γ has a subset A0 ⊆ A satisfying (a)-(c) in G(m). Monotonicity then implies (a)-(c) hold for A0 in all steps 0 ≤ i ≤ m, yielding Dm and so the lemma. For a given A ⊆ [n] with 2Γ ≤ |A| ≤ (log n)/7 · Γ, we form A0 ⊆ A as follows. First, let 0 | ≤ (13/2)|A|, we let B be A1 = A ∩ DA,log n , noting |A1 | ≤ 16ε−1 |A|/(log n). Next, as |DA,2 0 0 an arbitrary set (possible DA,2 itself) satisfying DA,2 ⊆ B ⊆ [n] \ A and Γ ≤ |B| ≤ (13/2)|A|, and then we let A2 = A ∩ DB,log n . Since Γ ≤ |B| ≤ (13/2)|A| < (log n) · Γ, it follows that |A2 | ≤ 16ε−1 |B|/(log n) ≤ 104ε−1 |A|/(log n). Finally, we let A0 = A \ (A1 ∪ A2 ), and so (a) follows by the bounds on |A1 | and |A2 |. We next show (c) holds: let v ∈ A0 . Since v ∈ / A1 , v has at most log n neighbors in A0 (which 0 may or may not have other neighbors in A ). So, it suffices to show that v has at most log n 0 , and neighbors z ∈ Nm (v) \ A satisfying |Nm (z) ∩ A0 | ≥ 2. Any such vertex z must lie in DA,2 therefore in B: since v ∈ / A2 , |Nm (v) ∩ B| ≤ log n and (c) follows. 0 Since A ∩ A1 = ∅ implies A0 induces a subgraph of maximum degree at most log n in G(m), it follows that A0 contains fewer than |A0 | · log n < Γ|A0 |/2 edges and fewer than |A0 | · log2 n < Γ|A0 |/2 pairs of vertices with a common neighbor lying inside A0 . To show (b) holds, it therefore suffices to show that at most 13Γ|A0 | pairs of vertices in A0 have a common neighbor outside of A0 . Since ∆(G(m)) ≤ Γ, the number of such pairs is at most   Γ X |Nm (z) ∩ A0 | X d 0 0 |) | − |D (|DA = 0 ,d A0 ,d+1 2 2 0

z∈[n]\A

d=2

Γ X

    d d−1 − + = 2 2 d=3   Γ 13|A0 | X d−1 + 13|A0 | ≤ 2 d 0 |DA 0 ,2 |

0 |DA 0 ,d |

d=3

≤ 13Γ|A0 |.

5.2

The proof itself

We now complete the proof of Lemma 1. We fix a step i, m/2 ≤ i ≤ m, a k-element set K ⊆ [n], and a vertex w ∈ [n] \ K, and we suppose K is w-uncovered in G(i). We assume that Ti , Di , and the bounds of Theorem 3 hold, as their collective failure probability (over all steps 0 ≤ i ≤ m) is o(1). It suffices to show that (12) holds, and the remainder of our argument is purely deterministic. First, since Di holds and k = C · Γ for some constant C > 56 by (11), we choose a subset 0 K ⊆ K satisfying properties (a)-(c) of Di , and we let   0  K K0 = (u, v) : uv ∈ \ U (i) . 2

17

As |K 0 | = (1 − o(1)) · k by (a), and as (b) holds and C > 28, it follows that   0    |K | 28 0 0 k 2 = Ω(k 2 ). |K | ≥ 2 − 14Γ|K | = (1 − o(1)) 1 − C 2

(19)

For each ordered pair of vertices (u, v) ∈ K0 , we also claim the following bounds: |Xu,v,`−3 (i)| ≥ (1 − o(1)) · x`−3 (t)S`−3 ≥ (1 − o(1)) ·

µ`−3 `−3 Γ · q(t) = Ω(Γ`−3 n−ε ). (` − 3)!

(20)

The first is implied by Theorem 3 and (5), the second as i ≥ m/2 implies t ≥ tmax /2, and thus x`−3 (t) ≥ (tmax )`−3 q(t)/(` − 3)!, and the final as q(t) ≥ n−ε by (5). We will remove a small number of extensions from the set Xu,v,`−3 (i) for each (u, v) ∈ K0 , and then argue that all but a negligible fraction of the remaining extensions count distinct open pairs lying in RK,w (i). Let D = {x ∈ [n] : |Ni (x) ∩ K 0 | ≥ 2}. For each ordered pair (u, v) ∈ K0 , we define w to consist of three ‘types’ of extensions: those which contain w the set of excluded extensions Yu,v in their image, those which count an open pair entirely contained in K 0 , and those extensions ϕ for which the first vertex ϕ1 lies in K 0 or has a neighbor in K 0 other than u. In the notation of Section 4.1, this is the set    [  [ w Φiu,v,`−3 (x; 1) . Φiu,v,`−3 (x; ` − 3) ∪ = Φiu,v,`−3 (w) ∪ Yu,v x∈K 0 ∪D

x∈K 0

w w := X bu,v We then define the set of remaining extensions X u,v,`−3 (i) \ Yu,v . w | is small relative to |X To show |Yu,v u,v,`−3 (i)|, we bound the number of extensions of each of the three types. First, by (17) we have |Φiu,v,`−3 (w)| ≤ (` log n)Γ−1 ·S`−3 ≤ (` log n)·Γ`−4 . Next, we note that if ` = 4, so ` − 3 = 1, the second set of extensions being excluded is contained in the third and requires no additional count. But if ` ≥ 5, then as |Φiu,v,`−3 (x; `−3)| ≤ 1 for all vertices x ∈ [n], S we have | x∈K 0 Φiu,v,`−3 (x; ` − 3)| ≤ k. For the third bound, we observe that if for some vertex x, Φiu,v,`−3 (x; 1) 6= ∅, then x is a neighbor of u. Since u ∈ K 0 and (c) holds, |Ni (u)∩(K 0 ∪D)| ≤ 2 log n, and for each vertex x ∈ Ni (u) ∩ (K 0 ∪ D) we may bound |Φiu,v,`−3 (x; 1)| ≤ Γ`−4 by (16). Combining these observations, and letting δ4` denote the Kronecker delta, we have w |Yu,v | ≤ (` log n)Γ`−4 + (1 − δ4` ) · k + (2 log n)Γ`−4 = o(Γ`−3 n−ε ),

(21)

w | = (1 − o(1)) · x bu,v as ε < 1/(` − 1), and thus |X `−3 (t)S`−3 by (20). S w , most of the remaining extenWe next show that after pruning the extensions in (u,v)∈K0 Yu,v sions count unique open pairs. To that end, we define the sets [ bw X = X and X = {(ϕ, θ) ∈ X × X : ϕ 6= θ, ϕ`−3 ϕ`−2 = θ`−3 θ`−2 } . u,v (u,v)∈K0

S w . As The set X is the set of all extensions considered above after removing those in (u,v)∈K0 Yu,v no extension in X contains w in its image, it follows that each extension ϕ ∈ X counts an open pair ϕ`−3 ϕ`−2 lying in RK,w (i). The set X consists of the ordered pairs of extensions in X which count the same open pair. Since |X| is at least the number of extensions ϕ lying in a pair in X, at least |X | − |X| extensions in X count unique open pairs, and so |RK,w (i)| ≥ |X | − |X|. 18

It remains to bound |X | and |X|: (19)-(21) imply |X | ≥ (1 − o(1)) · (x`−3 (t) · S`−3 ) · |K0 | = Ω(Γ`−3 n−ε k 2 ),

(22)

and so we will show that |X| = o(Γ`−3 n−ε k 2 ). Consider a pair of extensions (ϕ, θ) ∈ X ∗ . As the open pair ϕ`−3 ϕ`−2 = θ`−3 θ`−2 is not contained in K 0 , it follows that ϕ`−3 = θ`−3 ∈ [n] \ K 0 ,

ϕ`−2 = θ`−2 ∈ K 0 ,

{ϕ0 , θ0 } ⊆ K 0 ,

and, by (iv), ϕ0 6= θ0 as ϕ 6= θ. It then follows from (iv) that the pair (ϕ, θ) is uniquely determined by the four vertices θ0 , ϕ0 , ϕ`−3 = θ`−3 , and ϕ`−2 = θ`−2 . So, to bound |X|, we first select three vertices ϕ0 , θ0 , ϕ`−2 = θ`−2 lying in K 0 such that (ϕ0 , ϕ`−2 ), (θ0 , θ`−2 ) ∈ K0 , and we then bound the number of ways to select a fourth vertex z = ϕ`−3 = θ`−3 which can produce a pair (ϕ, θ) ∈ X. For any such vertex z, by (iv) there must be unique shortest-distance paths of length ` − 3 in G(i) from ϕ0 to z and from θ0 to z. Furthermore, as ϕ ∈ / Yϕw0 ,ϕ`−2 , Ni (ϕ1 ) ∩ K 0 = {ϕ0 } (since ϕ1 ∈ / D), and thus θ0 ϕ1 ∈ / E(i). These observations are illustrated in Figure 3. If ` = 4, so ` − 3 = 1, no vertex z can produce such a pair of extensions, as otherwise θ0 θ1 = θ0 ϕ1 ∈ / E(i), contradicting (i). That is, in the C4 -free process, X = ∅. Supposing instead that ` ≥ 5, we may apply Lemma 4 (with u = ϕ0 and x = θ0 ) to bound the number of choices of the fourth vertex z above by n8(`−1)ε Γ`−5 . As there are at most k 3 choices of the three vertices ϕ0 , θ0 , ϕ`−2 = θ`−2 , we conclude |X| ≤ k 3 (n8(`−1)ε Γ`−5 ) = C ·

n8(`−1)ε · Γ`−3 k 2 = o(Γ`−3 n−ε k 2 ), Γ

(23)

noting for the final bound that n8(`−1)ε /Γ < n−2ε , say, by (4). Finally, by (19), (20), (22), (23), and as C > 56, we obtain (12) for n sufficiently large as   28 1 (2tnp)`−3 k 2 |RK,w (i)| ≥ |X | − |X| ≥ (1 − o(1)) · x`−3 (t)S`−3 1 − k2 > · . C 2 (` − 3)!

6

The one-step changes

The remainder of our this paper is devoted to the proof of Theorem 3, which will follow from an application of the differential equations method in the next section. This application requires precise estimates on the expected one-step increase and decrease in the variables |Xu,v,j (i)|, as well as bounds on the maximum one-step increase and decrease. Estimating the expected one-step decrease will be a straightforward task, as an extension ϕ ∈ Xu,v,j (i) satisfies ϕ ∈ / Xu,v,j (i + 1) if conditions (ii), (iv) or (v) fail. Theorem 2 and Lemma 6 will be shown to yield a suitable estimate on the expected value of |Xu,v,j (i) \ Xu,v,j (i)|. To handle the expected one-step increase, though, we must take more care: conditions (i)-(v) ensure that any extension ϕ ∈ Xu,v,j (i + 1) \ Xu,v,j (i) must satisfy ϕ ∈ Xu,v,j−1 (i) and ei+1 = ϕj−1 ϕj . (This will be justified formally in the proof of Lemma 12 below.) However, it is possible that an extension ϕ ∈ Xu,v,j−1 (i) might not satisfy ϕ ∈ Xu,v,j (i + 1) on the event ei+1 = ϕj−1 ϕj . For example, the selection ei+1 = ϕj−1 ϕj might violate (ii) by closing 19

ϕ0 K0

ϕ1

ϕ2

ϕa−1 ϕa = θa

ϕ`−4 = θ`−4

ϕa+1 = θa+1

θ0 θ1

θ2

θa−1 ϕ`−3 = θ`−3

w

ϕ`−2 = θ`−2

Figure 3: Our view of a pair of extensions (ϕ, θ) ∈ X which count the same open pair ϕ`−3 ϕ`−2 = θ`−3 θ`−2 . (Here w is drawn outside of the images of ϕ and θ to emphasize that the shared open pair lies in RK,w (i).) Observe that once the paths from ϕ0 to ϕ`−3 and θ0 to θ`−3 intersect (at ϕa = θa ), they must coincide by (iv). Furthermore, the pair θ0 ϕ1 cannot form an edge in G(i), as ϕ1 ’s only neighbor in K 0 is ϕ0 by construction, and so a ≥ 2 must hold. one of the open pairs ϕj ϕj+1 , . . . , ϕ`−3 ϕ`−2 , or could violate (v) by resulting in disti+1 (u, ϕa ) ≤ a for one of the ‘free’ vertices ϕa , j + 1 ≤ a ≤ ` − 3. If j = 0, it could also yield uv ∈ U (i + 1), in which case Xu,v,j (i + 1) = Xu,v,j (i) by definition. The following lemma, analogous to Lemma 8.3 of [2], shows that these problems can only arise for a small fraction of the extensions in Xu,v,j−1 (i) relative to its scaling Sj−1 . Lemma 11 (Creation fidelity). Suppose n is suitably large, and let 0 ≤ i ≤ m and suppose Ti holds. Then for any index j, 0 ≤ j ≤ ` − 4, and any ordered pair (u, v) of vertices such that
uv ∈ [n] 2 \ U (i), the number of extensions ϕ ∈ Xu,v,j (i) for which the selection of ei+1 = ϕj ϕj+1 would not yield ϕ ∈ Xu,v,j+1 (i + 1) is at most n−8ε · Sj . The proof of Lemma 11 is given in Section 6.1. Bounds on the maximum one-step increase and decrease in these variables are provided in the following lemma, which we will prove in Section 6.2. Lemma 12 (Maximum increase and decrease). Provided n is suitably large, for all steps 0 ≤ i ≤ m − 1, indices 0 ≤ j ≤ ` − 3, and distinct vertices u, v ∈ [n], if Ti holds then max{ |Xu,v,j (i + 1) \ Xu,v,j (i)| , |Xu,v,j (i) \ Xu,v,j (i + 1)| } ≤ n−8ε · Sj .

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We mention that the bound on the maximum increase will follow easily from (18). The bounds on the maximum decrease will also follow through fairly simple estimates, except in the case j = ` − 3, where we appeal to an argument utilizing Lemma 5. We remark that this is the only part of our argument that requires that uv ∈ / E(i), i.e. that E(i) ⊆ U (i).

6.1

Creation fidelity: proof of Lemma 11

Throughout this section we fix a step i, 0 ≤ i ≤ m − 1, and assume the event Ti holds. We then fix an index j, 0 ≤ j ≤ ` − 4, and let u, v ∈ [n] be distinct vertices such that uv ∈ / U (i). 20

Consider an extension ϕ ∈ Xu,v,j (i). On the event ei+1 = ϕj ϕj+1 , ϕ trivially satisfies (i) and (iii) for membership in Xu,v,j+1 (i+1). We also claim it satisfies (iv), which is equivalent to the assertion that (u = ϕ0 , ϕ1 , . . . , ϕj+1 ) is a unique shortest-distance path in the graph G(i) + ϕj ϕj+1 . This follows as (ϕ0 , . . . , ϕj ) is a unique shortest path in G(i) and as disti (ϕ0 , ϕj+1 ) > j + 1 by (iv) and (v), respectively. So, if the selection of ei+1 = ϕj ϕj+1 does not result in ϕ ∈ Xu,v,j+1 (i + 1), at least one of the following holds: uv ∈ U (i + 1), or conditions (ii) or (v) fail. To prove Lemma 11, we will show in Claims 1-3 below that at most (n−8ε /3) · Sj extensions can produce each of these problematic situations. Bounding the number of extensions which fail condition (ii) is the most technical of these arguments, so we begin with the simpler estimates. Claim 1. The number of extensions ϕ ∈ Xu,v,j (i) for which uv ∈ U (i + 1) on the event ei+1 = ϕj ϕj+1 is at most (n−8ε /3) · Sj . Proof. Consider an extension ϕ ∈ Xu,v,j (i): since j ≤ ` − 4, v = ϕ`−2 is not an endpoint of ϕj ϕj+1 . Thus, if the selection ei+1 = ϕj ϕj+1 results in uv ∈ U (i + 1), the edge ϕj ϕj+1 must complete a path of length 2 between u and v. This implies j = 0 (since then u must be an endpoint of ϕj ϕj+1 ), S and that ϕ1 v ∈ E(i). Thus, we are bounding | w∈Ni (v) Φiu,v,0 (w; 1)|, the number of extensions ϕ ∈ Xu,v,0 (i) for which ϕ1 ∈ Ni (v). By (7), |Ni (v)| ≤ ∆(G(i)) ≤ Γ, and by (16), |Φiu,v,0 (w; 1)| ≤ n`−4 for all w ∈ Ni (v). This yields a bound of Γ · n`−4 = (Γ/n) · S0 such extensions. By (14), Γ/n = (log n)/Γ`−2 = n−(`−2)/(`−1)+o(1) , so by (4) we have Γ/n = o(n−8ε ), implying the claim for large n. Next, we bound the number of extensions ϕ ∈ Xu,v,j (i) for which condition (v) for membership in Xu,v,j+1 (i + 1) is violated on the event ei+1 = ϕj ϕj+1 . Observe that this condition depends on the ‘free’ vertices ϕj+2 , . . . , ϕ`−3 , and thus can only occur if j ≤ ` − 5, requiring ` ≥ 5. Claim 2. The number of extensions ϕ ∈ Xu,v,j (i) for which the selection of ei+1 = ϕj ϕj+1 results in disti+1 (u, ϕa ) ≤ a for some index a, j + 2 ≤ a ≤ ` − 3, is at most (n−8ε /3) · Sj . Proof. We assume that ` ≥ 5 and j ≤ ` − 5 throughout, as the bound is trivial otherwise. Fix an index a, j + 2 ≤ a ≤ ` − 3, and suppose that ϕ ∈ Xu,v,j (i) an extension for which the distance between u and ϕa is at most a in the graph G0 = G(i) + ϕj ϕj+1 . Let P be a shortest path u and ϕa in G0 , of length b ≤ a. Since disti (u, ϕa ) > a by (v), P must include the edge ϕj ϕj+1 . As (iv) and disti (u, ϕj+1 ) > j + 1 imply the path P 0 = (u = ϕ0 , ϕ1 , . . . , ϕj , ϕj+1 ) is a unique shortest path in G0 , P must be the union of P 0 and another path P 00 between ϕj+1 and ϕa and of length b − (j + 1) ≤ a − (j + 1) ≤ ` − 4, contained entirely in G(i). To bound the number of such extensions ϕ, we first select a path of length j starting at u in at most ∆(G(i))j ≤ Γj ways, and then we select the ‘free’ vertices in {ϕj+1 , . . . , ϕ`−3 } \ {ϕa } in at most n`−4−j ways. Finally, we select ϕa at distance at most ` − 4 from ϕj+1 , which can be done in at most Γ + Γ2 + · · · + Γ`−4 ≤ 2Γ`−4 ways. Allowing a to vary over its fewer than ` possible values gives an upper bound on the total number of such extensions of     2` log2 n 2`Γ`−4 j `−4−j `−4 j `−3−j Γ ·n · (2Γ ) · ` = Γ n ≤ · Sj = n−3/(`−1)+o(1) · Sj n Γ3 by (14). By (4), we have n−3/(`−1)+o(1) = o(n−8ε ), and the claim follows for large n. 21

Finally, we bound the number of extensions ϕ ∈ Xu,v,j (i) for which condition (ii) would be violated on the event ei+1 = ϕj ϕj+1 . Claim 3. The number of extensions ϕ ∈ Xu,v,j (i) for which ϕj ϕj+1 ∈ Cϕa ϕa+1 (i) for some index a, j + 1 ≤ a ≤ ` − 3, is at most (n−8ε /3) · Sj . Proof. For the special case ` = 4, so j = 0, we claim that as uv ∈ / U (i), there are no such extensions. To see this, if ϕ ∈ Xu,v,0 (i) satisfies ϕ0 ϕ1 ∈ Cϕ1 ϕ2 (i), then a copy of C4 exists in the graph G(i) + ϕ0 ϕ1 + ϕ1 ϕ2 , implying G(i) contains a path of length 2 between ϕ0 = u and ϕ2 = v. We therefore assume ` ≥ 5, and our argument proceeds by considering three cases, motivated as follows. If ϕ ∈ Xu,v,j (i) and ϕj ϕj+1 ∈ Cϕa ϕa+1 (i) for some index a, then a copy of C` exists in G(i) + ϕj ϕj+1 + ϕa ϕa+1 using both ϕj ϕj+1 and ϕa ϕa+1 as edges. If a > j + 1, then ϕj ϕj+1 and ϕa ϕa+1 are nonadjacent, so there exist ‘short’ paths in G(i) connecting the vertices ϕj and ϕj+1 to the vertices ϕa and ϕa+1 . In this case, ϕa is a free vertex of short distance to ϕj or to ϕj+1 , yielding a bound through an argument similar to that of Claim 2 above. If a = j + 1, so ϕj ϕj+1 and ϕa ϕa+1 are adjacent, a path of length ` − 2 exists in G(i) between ϕj and ϕa+1 . Our second case considers the situation where ϕa+1 = ϕj+2 is a free vertex, and a similar argument applies. Note that if ϕa+1 = ϕj+2 is a free vertex, then j ≤ ` − 5. This leaves only the case j = ` − 4 and a = ` − 3, where ϕa+1 = ϕ`−2 = v. In this case, a path of length ` − 2 exists in G(i) between v and ϕj = ϕ`−4 , and we obtain a suitable bound from an application of Lemma 5. Case 1: j < ` − 4 and a > j + 1. As noted above, if ϕ ∈ Xu,v,j (i) satisfies ϕj ϕj+1 ∈ Cϕa ϕa+1 (i), then G(i) contains two vertex-disjoint paths of total length ` − 2 connecting ϕj and ϕj+1 to ϕa and ϕa+1 . As each path has at least one edge, it follows that min{disti (ϕa , ϕj ) , disti (ϕa , ϕj+1 )} ≤ ` − 3. Thus, we may bound the number of choices of the vertices ϕ0 , . . . , ϕj above by Γj , and of the free vertices in {ϕj+1 , . . . , ϕ`−3 } \ {ϕa } above by n`−3−j−1 . Finally, there are at most 2(Γ + Γ2 + · · · + Γ`−3 ) ≤ 4Γ`−3 ways to select the free vertex ϕa within distance ` − 3 of ϕj or ϕj+1 , yielding the number of such extensions in this case is at most  `−3   `−3  4Γ 4Γ log n Γj n`−3−j−1 · (4Γ`−3 ) = Γj n`−3−j ≤ Sj n n by (14). Case 2: j < `−4 and a = j +1. In this case, if ϕj ϕj+1 ∈ Cϕa ϕa+1 (i) it follows that disti (ϕj , ϕa+1 = ϕj+2 ) ≤ ` − 2. Selecting, as above, the vertices in {ϕ1 , . . . , ϕ`−3 } \ {ϕj+2 } first, and then choosing ϕj+2 to be one of the at most Γ + Γ2 + · · · + Γ`−2 ≤ 2Γ`−2 vertices within distance ` − 2 of ϕj , we obtain the bound  `−2    2Γ 2 log2 n j `−3−j−1 `−2 j `−3−j Γ ·n · (2Γ ) = Γ n ≤ · Sj . n Γ Case 3: j = ` − 4 and a = ` − 3. Consider an extension ϕ ∈ Xu,v,`−4 (i) for which ϕ`−4 ϕ`−3 ∈ Cϕ`−3 v (i): as noted above, a path of length `−2 exists in G(i) between v and ϕ`−4 . As disti (u, ϕ`−4 ) = 22

` − 4, and as uv ∈ / U (i) implies disti (u, v) > 2 = (` − 2) − (` − 4), it follows from Lemma 5 (with x = v, d = ` − 2, and u and j as above) that the number of vertices z which could serve as ϕ`−4 in such an extension is at most n8(`−1)ε Γ`−5 . Furthermore, by (16), each such vertex z serves as ϕ`−4 in |Φiu,v,`−4 (z; ` − 4)| ≤ n extensions, yielding the upper bound n8(`−1)ε Γ`−5 · n ≤

n8(`−1)ε log n Γ

! S`−4 .

To complete the proof of the claim, if j ≤ ` − 5, then by allowing a to vary over fewer than ` values, Cases 1 and 2 yield the total number of such ‘bad’ extensions is less than   4 log2 n 2 log2 n 4(` + 1) log2 n `· + · S ≤ · Sj = n−1/(`−1)+o(1) · Sj , j Γ2 Γ Γ which suffices as n−1/(`−1)+o(1) = o(n−8ε ) by (4). If j = ` − 4, then the bound follows from Case 3, as (4) implies n8(`−1)ε (log n) · Γ−1 = n8(`−1)ε−1/(`−1)+o(1) = o(n−8ε ). This completes the proof of Claim 3, and thus of Lemma 11.

6.2

Maximum increase and decrease - Proof of Lemma 12

To prove Lemma 12, we again fix a step i, 0 ≤ i ≤ m−1, and assume Ti holds. We then fix an index j, 0 ≤ j ≤ ` − 3, and let u, v ∈ [n] be distinct vertices. As (24) is trivial on the event uv ∈ U (i + 1) (as then Xu,v,j (i + 1) = Xu,v,j (i)), we will further assume that uv ∈ / U (i), and that the the edge ei+1 is selected so that uv ∈ / U (i + 1). We begin by establishing (24) for the maximum increase. Since O(i + 1) ⊆ O(i) and since distances are nonincreasing as the process evolves, it follows that |Xu,v,0 (i + 1) \ Xu,v,0 (i)| = 0. We therefore assume that j > 0, and we will show next that Xu,v,j (i + 1) \ Xu,v,j (i) ⊆ Φiu,v,j−1 (ei+1 ). (Recall that for an open pair xy, Φiu,v,j (xy) is the set of extensions ϕ ∈ Xu,v,j (i) for which xy ∈ {ϕj ϕj+1 , . . . , ϕ`−3 ϕ`−2 }.) Let ϕ ∈ Xu,v,j (i + 1) \ Xu,v,j (i) be arbitrary. We first claim that ϕj−1 ϕj = ei+1 . Otherwise, {ϕ0 ϕ1 , . . . , ϕj−1 ϕj } ⊆ E(i) by (iii), and with this, conditions (i)-(v) for ϕ ∈ Xu,v,j (i + 1) imply ϕ ∈ Xu,v,j (i), a contradiction. It is routine to verify that ei+1 = ϕj−1 ϕj and ϕ ∈ Xu,v,j (i + 1) imply that conditions (i)-(iv) for ϕ ∈ Xu,v,j−1 (i) hold, and that disti (u, ϕa ) > a for j + 1 ≤ a ≤ ` − 3. To establish condition (v), we need only argue that disti (u, ϕj ) > j: this follows as (u = ϕ0 , . . . , ϕj−1 , ϕj ) is a unique shortest-distance path in G(i + 1), and ϕj−1 ϕj = ei+1 ∈ / E(i). i Thus, ϕ ∈ Xu,v,j−1 (i) and ϕj−1 ϕj = ei+1 , so Xu,v,j (i + 1) \ Xu,v,j (i) ⊆ Φu,v,j−1 (ei+1 ) as claimed. An application of (18) then yields     2` log n 2` log n · Sj−1 = · Sj , |Xu,v,j (i + 1) \ Xu,v,j (i)| ≤ n np establishing the first bound of (24) for large n as (2` log n)/(np) = n−1/(`−1)+o(1) = o(n−8ε ) by (4). Turning to the maximum decrease, if ϕ ∈ Xu,v,j (i) \ Xu,v,j (i + 1), then ϕ satisfies (i) and (iii) for membership in Xu,v,j (i + 1), and so one of conditions (ii), (iv) or (v) must fail. In Claim 4 below, we show that at most (n−8ε /2) · Sj extensions ϕ ∈ Xu,v,j (i) can violate (iv) or (v) (in G(i + 1)). 23

Then, in Claim 5, we show at most (n−8ε /2) · Sj extensions ϕ ∈ Xu,v,j (i) \ Xu,v,j (i + 1) can violate (ii). This argument considers two cases: 0 ≤ j ≤ ` − 4, and j = ` − 3. In the former case, the bound follows by considering the maximum number of extensions in Xu,v,j (i) which include an open pair from O(i) \ O(i + 1) = Cei+1 (i) ∪ {ei+1 }. However, this bounding approach is too weak to suffice if j = ` − 3, as S`−3 < p−1 < |Cei+1 (i)| for large n and most steps i being considered, so additional care will be necessary. Claim 4. The number of extensions ϕ ∈ Xu,v,j (i) \ Xu,v,j (i + 1) for which conditions (iv) or (v) for membership in Xu,v,j (i + 1) fail is at most (n−8ε /2) · Sj . Proof. If ϕ ∈ Xu,v,j (i) \ Xu,v,j (i + 1) and conditions (iv) or (v) fail, then ei+1 must create a path of length at most a between u and ϕa , for some index 1 ≤ a ≤ ` − 3. Letting x and y denote the endpoints of ei+1 , suppose first that 1 ≤ a ≤ j: then min{disti (x, ϕa ) , disti (y, ϕa )} ≤ a − 1. By (7), there are fewer than 4Γa−1 vertices z within distance a − 1 of x or of y in G(i), and by (16) each such vertex z serves as ϕa in at most (log n)Γ−a · Sj extensions ϕ ∈ Xu,v,j (i). If j + 1 ≤ a ≤ ` − 3, a nearly identical argument shows at most 4Γa−1 vertices z which could serve as ϕa in such an extension, and, by (16), each can do so in at most (log n)n−1 · Sj extensions ϕ ∈ Xu,v,j (i). Thus, the total number of extensions in Xu,v,j (i) which violate conditions (iv) or (v) in G(i + 1) is at most j X

4Γ

a−1

−a

· (log n)Γ

a=1

· Sj +

`−3 X

4Γa−1 · (log n)n−1 · Sj < (4` log n)/Γ · Sj ,

a=j+1

which suffices as (4` log n)/Γ = n−1/(`−1)+o(1) = o(n−8ε ) by (4). Claim 5. The number of extensions ϕ ∈ Xu,v,j (i) \ Xu,v,j (i + 1) for which condition (ii) fails for membership in Xu,v,j (i + 1) is at most (n−8ε /2) · Sj . Proof. As mentioned, we separate the cases 0 ≤ j ≤ ` − 4 and j = ` − 3. Case 1: 0 ≤ j ≤ ` − 4. For each extension ϕ ∈ Xu,v,j (i) \ Xu,v,j (i + 1) for which (ii) fails, we have {ϕj ϕj+1 , . . . , ϕ`−3 ϕ`−2 }∩(Cei+1 (i)∪{ei+1 }) 6= ∅. For each pair wz ∈ Cei+1 (i)∪{ei+1 }, by (18) there are |Φiu,v,j (wz)| ≤ (2` log n)/n · Sj extensions ϕ ∈ Xu,v,j (i) with wz ∈ {ϕj ϕj+1 , . . . , ϕ`−3 ϕ`−2 }. Furthermore, we claim |Cei+1 (i) ∪ {ei+1 }| ≤ (log n) · p−1 for large n, which follows from (8) as f (t) < nε < se by (5) and as c(t) = o(log n), noted in Section 4. Thus, the total number of extensions in Xu,v,j (i) for which (ii) fails in G(i + 1) is at most (log n)p−1 ·

2` log n 2` log2 n · Sj = · Sj = n−1/(`−1)+o(1) · Sj , n np

which more than suffices. Case 2: j = ` − 3. In this case, for each extension ϕ ∈ Xu,v,`−3 (i) \ Xu,v,`−3 (i + 1) failing (ii), we have ϕ`−3 v ∈ Cei+1 (i) ∪ {ei+1 }. As there is at most one such extension with ϕ`−3 v = ei+1 (by 24

Lemma 3), we focus on bounding the remainder. Let B denote this set, i.e. the set of all extensions ϕ ∈ Xu,v,`−3 (i) \ Xu,v,`−3 (i + 1) such that ϕ`−3 v ∈ Cei+1 (i), and let x and y denote the endpoints of ei+1 . For each extension ϕ ∈ B, select a copy C`,ϕ of C` in the graph G(i) + ei+1 + ϕ`−3 v using ∗ denote the subgraph of C the edges ei+1 and ϕ`−3 v, and let C`,ϕ `,ϕ formed by deleting the edges ∗ ei+1 and ϕ`−3 v. Note that C`,ϕ is a subgraph of G(i) consisting of two vertex-disjoint paths which connect x and y to ϕ`−3 and v (or vice-versa). Let Bx ⊆ B consist of those extensions ϕ ∈ B for ∗ consists of a path between x and ϕ which C`,ϕ `−3 (and a path between y and v), and let By = B\Bx . We bound |Bx |, noting by symmetry our arguments will also bound |By |. ∗ consists of the isolated vertex v = x and a path of First, if x = v, then for every ϕ ∈ B, C`,ϕ length ` − 2 between y and ϕ`−3 , implying Bx = ∅. We therefore assume x 6= v, and we partition Bx into (possibly empty) sets as follows. For each length d, 0 ≤ d ≤ ` − 2, let Bxd ⊆ Bx consist of ∗ has length d. those extensions ϕ for which the path between x and ϕ`−3 in C`,ϕ ∗ between y and v has length at least one and If y 6= v, then for each ϕ ∈ Bx , the path in C`,ϕ S d so the path between x and ϕ`−3 has length at most ` − 3. Consequently, Bx = `−3 d=0 Bx . For each d d d d length d, we claim that |Bx | ≤ Γ : this follows as there are at most ∆(G(i)) ≤ Γ vertices z at the end of a path of length d from x in G(i), and, by Lemma 3, each vertex z serves as ϕ`−3 in at most S d `−3 , bounding one extension ϕ ∈ Xu,v,`−3 (i). This suffices to bound | `−4 d=0 Bx |, but as S`−3 < Γ |Bx`−3 | requires a finer argument. ∗ To bound |Bx`−3 | we apply Lemma 5, assuming Bx`−3 6= ∅. Let ϕ ∈ Bx`−3 , observing that C`,ϕ must contain the edge yv (and thus so does E(i)). Since ei+1 = xy, disti+1 (x, v) ≤ 2, but by our choice of ei+1 we have disti+1 (u, v) > 2 (else uv ∈ U (i + 1)), so u 6= x, i.e., disti (u, x) > 0. Applying Lemma 5 (with u, x, j = ` − 3 as above and d = ` − 3) yields at most n8(`−1)ε · Γ`−4 vertices z can serve as ϕ`−3 in some extension ϕ ∈ Bx`−3 , producing the same bound on |Bx`−3 |. Combining these observations and applying (14) yields that if y 6= v, |Bx | =

`−4 X

|Bxd | + |Bx`−3 | ≤ (2 + n8(`−1)ε ) · Γ`−4 <

d=0

2n8(`−1)ε log n · S`−3 . Γ

If y = v, then for every ϕ ∈ Bx the path between x and ϕ`−3 has length ` − 2 (so Bx = Bx`−2 ). We apply Lemma 5 again, taking u, x, j = `−3 as above and d = `−2. This requires disti (u, x) ≥ 2, which holds as otherwise we would have disti+1 (u, y = v) ≤ 2. The application of Lemma 5 then produces the bound |Bx | = |Bx`−2 | ≤ n8(`−1)ε · Γ`−4 < (2n8(`−1)ε (log n)/Γ) · S`−3 by (14). It follows that the total number of extensions ϕ ∈ Xu,v,`−3 (i) \ Xu,v,`−3 (i + 1) for which (ii) fails is at most 5n8(`−1)ε log n 1 + |Bx | + |By | ≤ · S`−3 = n8(`−1)ε−1/(`−1)+o(1) · S`−3 , Γ bounding 1 ≤ (n8(`−1)ε log n/Γ) · S`−3 , which suffices as n8(`−1)ε−1/(`−1)+o(1) = o(n−8ε ) by (4).

25

7 7.1

Proof of Theorem 3 The differential equations method

To prove Theorem 3, we appeal to a variant of Lemma 7.3 from [2] given by Warnke in [16]. We mention the preceding results suffice to track our variables with slightly different error terms using Lemma 7.3 of [2], but to do so requires accounting for local stopping times and small initial error (see [9] or [10]), which are absent from the statement in [2]. In addition to the explicit inclusion of such features, Warnke’s lemma also introduces additional error parameters which simplify the verification of certain technical conditions. We next give a short setup for this lemma, similar to that given in [2] for Lemma 7.3, and refer the reader to [16] for a more in-depth discussion. Suppose we have a stochastic graph process, parameterized by n, and positive parameters m = m(n) and s = s(n). Let C be a set of ‘configurations’, in our case ‘positions’ in the graph, and let V be a set of ‘types’ of variables. Suppose, for each configuration-type pair σ ∈ C ×V, we have a sequence of random variables Xσ (i), i = 0, . . . , m, measurable with respect to a common filtration Fi determined by the process. We suppose further that Xσ (i + 1) − Xσ (i) = Yσ+ (i) − Yσ− (i), where Yσ± (i) are nonnegative random variables measurable with respect to Fi+1 , and that for each configuration Σ ∈ C we have a local ‘bad’ event Bi (Σ), after which we ignore the variables corresponding to Σ. We relate these sequences to functions on [0, ∞) by setting t = t(i) = i/s. We wish to show, for some continuous functions xσ (t) and size-scalings Sσ , that an approximation of the form Xσ (i) ≈ xσ (t)Sσ holds for all steps i ≤ m and all σ = (Σ, j) ∈ C × V for which Σ is not being ignored. To do so, we show, in part, that for each such i and σ, conditioned on the desired approximation holding for all previous steps, E (Yσ± (i)) ≈ yσ± (t) · Sσ /s, where the yσ± are smooth, nonnegative functions satisfying x0σ = yσ+ −yσ− , and that |Yσ± (i)| is suitably smaller than Sσ . We remark that this separate treatment of the increase and decrease in the variables, which is not typical in most applications of the differential equations method, is a key element in the proof of this lemma. In our application, as defined in (1) and (2) we take m = µ(log n)1/(`−1) · n`/(`−1) and s = n2 p = n`/(`−1) . We define our set of ‘configurations’ C and our set of variable ‘types’ V as    [n] C = (u, v) : uv ∈ and V = {0, 1 . . . , ` − 3}. 2 That is, C is the set of ordered pairs of distinct vertices, and V is the set of indices j of the variables |Xu,v,j (i)| we will track. We let F0 , F1 , . . . denote the natural filtration determined by the first i steps of the process (see [1], for example). For each configuration Σ = (u, v) ∈ C, we define the local ‘bad’ event Bi (Σ) = Bi ((u, v)) to be the event uv ∈ U (i). For each pair σ = ((u, v), j) ∈ C × V, we take as our random variable, continuous approximation, and size-scaling, respectively, Xσ (i) = |Xu,v,j (i)|,

xσ (t) = xj (t),

and

Sσ = Sj .

We take as the error function and error scaling, respectively, fσ (t) = fj (t) := f (t)q(t)−j ,

and

sσ = n3ε .

We let the high probability event Hi = Ti , defined in Theorem 2, and we take the (technical) parameter uσ = nε (for all σ ∈ C × V). We defer discussion of the remaining variables and 26

parameters until after the statement, but applying the lemma with these values will directly yield that the estimates of Theorem 3 hold on the event Em of Lemma 13, and, further, Pr ((Em )c ) ≤ 4 exp(−nε ) + Pr (¬Tm ) = o(1), proving Theorem 3. Lemma 13 (Differential Equations Method, [16] Lemma 5.3). Suppose that m = m(n) and s = s(n) are positive parameters. Let C = C(n) and V = V(n) be sets. For every 0 ≤ i ≤ m set t = t(i) := i/s. Suppose we have a filtration F0 ⊆ F1 ⊆ · · · and random variables Xσ (i) and Yσ± (i) which satisfy the following conditions. Assume that for all σ ∈ C ×V the random variables Xσ (i) are non-negative and Fi -measurable for all 0 ≤ i ≤ m, and that for all 0 ≤ i < m the random variables Yσ± (i) are non-negative, Fi+1 -measurable and satisfy Xσ (i + 1) − Xσ (i) = Yσ+ (i) − Yσ− (i) .

(25)

Furthermore, suppose that for all 0 ≤ i ≤ m and Σ ∈ C we have an event Bi (Σ) ∈ Fi . Then, for S all 0 ≤ i ≤ m we define B≤i (Σ) := 0≤j≤i Bj (Σ). In addition, suppose that for each σ ∈ C × V we have positive parameters uσ = uσ (n), λσ = λσ (n), βσ = βσ (n), τσ = τσ (n), sσ = sσ (n) and Sσ = Sσ (n), as well as functions xσ (t) and fσ (t) that are smooth and non-negative for t ≥ 0. For all 0 ≤ i∗ ≤ m and Σ ∈ C, let Gi∗ (Σ) denote the event that for every 0 ≤ i ≤ i∗ and σ = (Σ, j) with j ∈ V we have   fσ (t) Xσ (i) = xσ (t) ± Sσ . (26) sσ Next, for all 0 ≤ i∗ ≤ m let Ei∗ denote the event that for every 0 ≤ i ≤ i∗ and Σ ∈ C the event B≤i−1 (Σ) ∨ Gi (Σ) holds. Moreover, assume that we have an event Hi ∈ Fi for all 0 ≤ i ≤ m with Hi+1 ⊆ Hi for all 0 ≤ i < m. Finally, suppose that the following conditions hold: 1. (Trend hypothesis) For all 0 ≤ i < m and σ = (Σ, j) ∈ C × V, whenever Ei ∩ B≤i (Σ)c ∩ Hi holds we have  
hσ (t) Sσ ± ± , (27) E Yσ (i) | Fi = yσ (t) ± sσ s where yσ± (t) and hσ (t) are smooth non-negative functions such that Z t 0 + − xσ (t) = yσ (t) − yσ (t) and fσ (t) ≥ 2 hσ (τ ) dτ + βσ .

(28)

0

2. (Boundedness hypothesis) For all 0 ≤ i < m and σ = (Σ, j) ∈ C × V, whenever Ei ∧ ¬B≤i (Σ) ∧ Hi holds we have β2 Sσ Yσ± (i) ≤ 2 σ · . (29) sσ λσ τσ uσ 3. (Initial conditions) For all σ ∈ C × V we have   βσ Xσ (0) = xσ (0) ± Sσ . 3sσ

(30)

4. (Bounded number of configurations and variables) We have max {|C|, |V|} ≤ min euσ . σ∈C×V

27

(31)

5. (Additional technical assumptions) For all σ ∈ C × V we have ( )      τ  sσ λσ 2 9sσ λσ βσ σ , s
(32) (33)

0

0≤t≤m/s

Z hσ (0) ≤ sσ λσ

m/s

and

|h0σ (t)| dt ≤ sσ λσ .

(34)

0

Then we have Pr ((Em )c ∩ Hm ) ≤ 4 max e−uσ . σ∈C×V

7.2

Trajectory verification

In addition to the values stated above, for each σ = ((u, v), j) ∈ C × V, let Yσ− (i) = |Xu,v,j (i) \ Xu,v,j (i + 1)|,

Yσ+ (i) = |Xu,v,j (i + 1) \ Xu,v,j (i)|, yσ+ (t) = yj+ (t) :=

2xj−1 (t) , and q(t)

yσ− = yj− (t) :=

(` − 2 − j)c(t)xj (t) , q(t)

where we take x−1 (t) ≡ 0 for all t. We note that (25) is immediate. Given the error function `−1 fσ (t) = fj (t) = f (t)q(t)−j , and recalling f (t) = eW (t +t) , we define   fj0 (t) (` − 1)(W + j2`−1 )t`−2 + W hσ (t) = hj (t) := = fj (t). (35) 2 2 Finally, we define the remaining parameters to be λσ = log2 n, 7.2.1

βσ = 1,

and

τσ = log n.

Trend hypothesis

We start our verification with the trend hypothesis, and fixing a step i, 0 ≤ i < m, and a configuration-type pair σ = ((u, v) , j) ∈ C × V. We assume the event Ei ∩ [B≤i ((u, v))]c ∩ Hi holds. Note that as Ei holds, our definitions and parameters above yield that every estimate asserted by Theorem 3 holds through step i, and as [B≤i ((u, v))]c holds, uv ∈ / U (i). + We by verifying (27) for E (Yσ (i)|Fi ). This is trivial if j = 0, as then yσ+ (t) = y0+ (t) = 2x−1 (t)/q(t) ≡ 0 and Yσ (i) = |Xu,v,0 (i + 1) \ Xu,v,0 (i)| = 0 since no new open pairs are created in any step. Supposing instead that j > 0, then as was shown in the proof of Lemma 12, each ∗ extension ϕ ∈ Xu,v,j (i + 1) \ Xu,v,j (i) satisfies ϕ ∈ Xu,v,j−1 (i) and ei+1 = ϕj−1 ϕj . Let Xu,v,j−1 (i) ⊆ Xu,v,j−1 (i) denote the set of extensions ϕ ∈ Xu,v,j−1 (i) for which ϕ ∈ Xu,v,j (i + 1) holds on the event ei+1 = ϕj−1 ϕj . Since ei+1 is chosen uniformly at random from O(i), this latter event occurs with probability 1/Q(i), and linearity of expectation yields E

Yσ+ (i)




| Fi =

X ∗ ϕ∈Xu,v,j−1 (i)

28

∗ |Xu,v,j−1 (i)| 1 = . Q(i) Q(i)

Noting Hi = Ti holds, we estimate Q(i)−1 : by (5), max{q(t)−1 , f (t)} < nε , so by (6)         nε 1 n2 nε q(t) 2 nε 3nε q(t) 2 Q(i) = 1 ± q(t) ± 1± ⊆ 1± ·n ⊆ 1± ·n , se se 2 se se 2 se 2 the last containment following from Lemma 2 as nε /se = o(1) by (5). Lemma 2 then yields     2 4nε 1 2 8n2ε 1 1 = 1± · 2 ⊆ ± . Q(i) se q(t) n q(t) se n2

(36)

∗ Since Hi = Ti holds, uv ∈ / U (i), and j −1 ≤ `−4, Lemma 11 yields |Xu,v,j−1 (i)| = |Xu,v,j−1 (i)|± · Sj−1 . Applying (13) (i.e., (26)) to estimate Xu,v,j−1 (i), it follows from (36) that    
fj−1 (t) 2 8n2ε 1 + −8ε Sj−1 · E Yσ (i) | Fi = xj−1 (t) ± ±n ± . n3ε q(t) se n2

n−8ε

Since fj−1 (t) = f (t)q(t)−(j−1) ≥ 1, n−8ε < fj−1 (t)/n3ε follows, and so   
2fj−1 (t) 2 8n2ε Sj−1 + ± E Yσ (i) | Fi = xj−1 (t) ± n3ε q(t) se n2     8xj−1 (t)n5ε 16fj−1 (t)n2ε Sj 1 + −1 ⊆ yσ (t) ± 4q(t) fj−1 (t) + + se se n3ε s  
1 Sj + −1 . ⊆ yσ (t) ± 5q(t) fj−1 (t) 3ε n s The first containment above was obtained by expanding, as Sj−1 /n2 = Sj /(n2 p) = Sj /s and yσ+ (t) = yj+ (t) = 2xj−1 (t)/q(t). The second containment follows for large n as q(t)fj−1 (t) ≥ 1, but both 8xj−1 (t)n5ε /se = o(1) and 8fj−1 (t)n2ε /se = o(1), by (5) and as xj−1 (t) = o(log n). Since n3ε = sσ and Sj = Sσ , to establish (27) for the increase, it only remains to argue that 5q(t)−1 fj−1 (t) < hσ (t) = hj (t). This follows as 5q(t)−1 fj−1 (t) = 5fj (t) by definition, while hj (t) > (W/2) · fj (t) > 5 · fj (t) by (35) and (4). Thus, E (Yσ+ (i)|Fi ) satisfies (27). We turn now to establishing (27) for E (Yσ− (i)|Fi ). For each extension ϕ ∈ Xu,v,j (i), we slightly P abuse notation and let pϕ = Pr (ϕ ∈ / Xu,v,j (i + 1)), so E (Yσ− (i)|Fi ) = ϕ∈Xu,v,j (i) pϕ , and we first focus on estimating pϕ for each such extension ϕ. Let ϕ ∈ Xu,v,j (i) be arbitrary: then ϕ ∈ / Xu,v,j (i + 1) if and only if uv ∈ / U (i + 1) and at least one of conditions (ii), (iv) or (v) fails. The failure of condition (ii) is equivalent to the assertion that ϕa ϕa+1 ∈ O(i + 1) \ O(i) = Cei+1 ∪ {ei+1 } for some index a, j ≤ a ≤ ` − 3. By symmetry, this is the event `−3 [ ei+1 ∈ (Cϕa ϕa+1 (i) ∪ {ϕa ϕa+1 }), a=j

and we will argue that this event yields the dominant contribution to pϕ . Now, the failure of conditions (iv) or (v) is the event that the edge ei+1 creates a (new) path of length at most a from u to ϕa for some index a, 1 ≤ a ≤ ` − 3. By Lemma 6, the number of open P a−1 < 4`Γ`−4 , say, pairs in O(i) whose selection as ei+1 produces this result is at most `−3 a=1 2`Γ for large n. Furthermore, the number of open pairs whose selection as ei+1 results in uv ∈ U (i + 1) 29

(i.e., completes a path of length at most 2 between u and v) is at most 2∆(G(i))) + 1 < 3Γ, say. We bound the total number of open pairs considered here by 4`Γ`−4 + 3Γ < 4Γ`−3 < p−1 /se for large n, the last inequality following from (14) and as (log n)se = n1/(2`)−ε+o(1) < Γ for large n. S To estimate | `−3 a=j (Cϕa ϕa+1 (i) ∪ {ϕa ϕa+1 })|, we apply (8) to estimate |Cϕa ϕa+1 (i)| for each index j ≤ a ≤ ` − 3, and (9) to bound the fewer than `2 intersections |Cϕa ϕa+1 (i) ∩ Cϕb ϕb+1 (i)| for j ≤ a < b ≤ ` − 3. We may also bound |{ϕj ϕj+1 , . . . , ϕ`−3 ϕ`−2 }| = (` − 2 − j) < p−1 /se : doing so, and incorporating the estimates of the previous paragraph yields      f (t) − 1 4(` − 1) p−1 `2 p−1 2p−1 1 pϕ = (` − 2 − j) 1 ± c(t) ± · ± 1/` ± , se se 2 se Q(i) n      nε 2`2 4 p−1 1 nε c(t) ± ± 1/` ± · , ⊆ (` − 2 − j) 1 ± se se se 2 Q(i) n   c(t)nε + nε + n2ε /se + 2`2 + 4 p−1 1 ⊆ (` − 2 − j)c(t) ± · , se 2 Q(i)   n2ε p−1 1 · . ⊆ (` − 2 − j)c(t) ± se 2 Q(i) Here we note that in the penultimate containment we bounded n−1/` < s−1 e , and in the final we 2ε 2ε appealed to the fact that c(t) = o(log n) and n /se = ω(n ). Applying (13) to estimate |Xu,v,j (i)| and (36) to estimate Q(i)−1 , for large n we have      
fj (t) n2ε p−1 1 4n2ε 2 − E Yσ (i) | Fi = xj (t) ± 3ε Sj · (` − 2 − j) · c(t) ± · ± n se 2 q(t) se n2    fj (t) (` − 2 − j) · c(t) 2n3ε Sj ± ⊆ xj (t) ± 3ε n q(t) se s     6ε (` − 2 − j)fj (t)c(t) 2xj (t)n Sj 2fj (t)n3ε 1 − ⊆ yσ (t) ± + + . 3ε q(t) se se n s The first containment above appeals to the bounds q(t)−1 < nε , c(t) = o(nε ), and 4n4ε /se = o(n3ε ) (by (5)), while the second follows after expanding as yσ− (t) = yj− (t) = (` − 2 − j)c(t)xj (t)/q(t). By (3) and by (4), (` − 2 − j)fj (t)c(t)/q(t) ≤ (` − 1)2 2`−1 t`−2 · fj (t) < (` − 1)W t`−2 /2 · fj (t), and as se > n7ε , fj (t) < nε , and xj (t) = o(log n), xj (t)n6ε /se + fj (t)n3ε /se < W/2 for large n. It
therefore follows from (35) that E (Yσ− (i) | Fi ) = yσ− (t) ± hj (t)/n3ε · Sj /s, and thus (27) holds. Finally, to show (28) holds, the equality follows from our selection of yj+ (t), yj− (t) and routine differentiation, and the inequality follows as hj = fj0 /2 and fj (0) = 1 = βσ . This completes the verification of the trend hypothesis. 7.2.2

Boundedness hypotheses

To verify the boundedness hypothesis, we again fix a step 0 ≤ i < m, a configuration-type pair σ = ((u, v) , j) ∈ C × V, and we assume the event Ei ∩ [B≤i ((u, v))]c ∩ Hi holds. To establish (29), 30

we must show the pair of bounds Yσ± (i) ≤

βσ2 2 sσ λσ τσ

·

Sj Sσ = 7ε . uσ n log3 n

Since Yσ+ (i) = |Xu,v,j (i + 1) \ Xu,v,j (i)|, Yσ− (i) = |Xu,v,j (i) \ Xu,v,j (i + 1)| and Hi = Ti holds, Lemma 12 yields Yσ± (i) < Sj /n8ε for large n, which suffices. 7.2.3

Initial conditions and bounded number of variables

Starting with the initial conditions, consider an arbitrary configuration-type pair σ = ((u, v) , j) ∈ C × V. If 1 ≤ j ≤ ` − 3, then xj (0) = 0, which, as E(0) = ∅, implies (30). If, instead, j = 0, then x0 (0) = 1, and |Xu,v,0 (0)| = (n − 2)`−3 = n`−3 − O(n`−4 ): βσ n`−3−3ε · Sσ = = ω(n`−4 ) 3sσ 3 since ε < 31 by (4), (30) follows for large n. And as uσ = nε and max{|C|, |V|} ≤ n2 , (31) is immediate. 7.2.4

Additional technical assumptions

The verification of (32)-(34) is routine but included for completeness. The first inequality in (32) becomes s ≥ max{15n7ε log5 n, 9n3ε log2 n}, which follows as s = n2 p ≥ n ≥ n8ε . The second inequality follows as s < m = o(log n)s and sσ , λσ ≥ 1. For (33), the first inequality follows from xj−1 (t) ≤ xj−1 (t)/q(t) = o(log n) and c(t)/q(t) = o(log n). The second follows from the observation that, for 0 ≤ j ≤ ` − 3, xj (t) = Atj · exp(−Bt`−1 ) for positive constants A and B depending only on j and `: it follows from routine differentiation that we can bound |x00j (t)| above by a function of the form F (t) · exp(−Bt`−1 ), where F (t) is a polynomial in t with positive coefficients. This is easily seen to have a bounded integral over R m/s [0, ∞), yielding 0 |x00j (t)| dt = O(1), and we note λσ = ω(1). Turning now to (34), by (35), for each configuration-type pair σ = (Σ, j) ∈ C × V, we have hσ (0) = hj (0) = W/2, implying the first inequality. From (35), hj (t) is the product of a polynomial with positive coefficients and fj (t) (which is strictly increasing). This implies first that |h0j (t)| = R m/s h0j (t), and therefore 0 |h0j (t)| dt = hj (tmax ) − W/2, and secondly that hj (tmax ) ≤ o(nε ) · n2ε = o(sσ λσ ), recalling that |F (t)| = o(nε ) on [0, tmax ] for any fixed polynomial F (t). This completes the verification of Lemma 13, and thus the proof of Theorem 3.

Acknowledgements The author wishes to thank the anonymous referees for numerous helpful corrections and suggestions.

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