Statement of Research Interests Michael E. Picollelli My current research interests are varied, with a focus on extremal and probabilistic combinatorics, especially problems in extremal graph theory and random graph theory.

1

Random Graphs with a Fixed Degree Sequence

Random graph models have received considerable attention since the initial work of Erd˝os and R´enyi (see [8] or [19], for example). In recent years, this attention has focused on models of sparse random graphs, with the aim of applying these to study large real-world networks with properties that are known to differ from the Erd˝os-R´enyi model (see [30], for example). One such model of interest to me is the Gz model, which produces graphs with a fixed degree sequence as follows. Let R+ denote the nonnegative reals, let ∆ be a positive integer, and let z ∈ (R+ )∆ with z∆ > 0. Finally, let n be a large integer. The Gz model produces graphs chosen uniformly at random from those with bzi nc or dzi ne vertices of degree i (the nearest-integer rounding is chosen to preserve parity). This is a generalization of the uniform model for random ∆-regular graphs on n vertices, which correspond to the selection 0 = z1 = · · · = z∆−1 and z∆ = 1. (See [10] for a related model in which the expected degree of a vertex is fixed, rather than the actual degree.) In mathematical epidemiology, random graphs are used to model the spread of disease on a network in which vertices (nodes) present individuals, and edges represent connections along which disease is potentially likely to spread. A classical example of this is the SIR model, due to Kermack and McKendrick [24]. In this infection model, nodes can be susceptible to infection (S), infected (I), or recovered (R). An infected node recovers completely according to an exponential distribution with parameter ρ, and while infected it transmits infections to its neighbors independently according to an exponential distribution with parameter λ. An infection begins with a single infected node, and ends when no nodes are actively infecting. A recent question [40] asks for a characterization of the likely evolution of such an SIR epidemic on Gz over time, as n → ∞. By combining techniques from branching process theory with the more recently developed differential equations method for randomized algorithm analysis [44], Tom Bohman and I have found that the evolution of the degree sequence over the life of such an epidemic is highly concentrated around the solution of an associated system of ordinary differential equations [7]. Our techniques also yield the limiting probability of a linear-sized epidemic, as well as the likely fraction of nodes infected by such an epidemic and the duration of time the number of actively infecting nodes lies above a linear threshold. These results are related to some of the earliest results on the Gz model due to Molloy and Reed, who determined conditions on z for which a giant (linear-sized) component is likely to exist [27], as well as the likely asymptotic size [28] of the giant component. Further results have been found on the size of the largest components in the critical phase [21], as well as in the subcritical phase of a power-law degree distribution [38] (see also [18]). There are several other natural questions about the Gz model which remain open. To simplify our statements, we say a sequence of events An occurs with high probability if limn→∞ P [An ] = 1. 1

Problem 1 For which z does Gz have a perfect matching w.h.p.? Bohman and Frieze [5] obtained partial results on this problem by analyzing the KarpSipser algorithm on Gz for log-concave distributions z. They conjectured that Gz has a perfect matching whenever z1 = z2 = 0, i.e. minimum degree 3, but I (and, independently, Mike Molloy - see [5]) have produced counterexamples for every minimum degree δ ≥ 3 for which Gz does not have even an almost-perfect matching, i.e. a matching of size (1 − o(1))|V (Gz )|/2. The question of whether perfect matchings exist invariably leads to that of whether Hamilton cycles exist. Problem 2 For which z is Gz Hamiltonian w.h.p.? In the case of random ∆-regular graphs (recalling that this is Gz where z has the form (0, . . . , 0, 1)), this question is resolved: Wormald and Robinson [46] showed that almost all random regular graphs are Hamiltonian. However, for general choices of z, little is known. For example, Wormald conjectures that random graphs with half of the vertices of degree three and half of degree four are Hamiltonian. Conjecture 1 (Wormald [45]) G(0,0,1,1) is Hamiltonian w.h.p.. As in the case of perfect matchings, a minimum degree condition alone is insufficient to guarantee Hamiltonicity (w.h.p.) - in particular, the counterexamples to the conjecture of Bohman and Frieze above also yield counterexamples to a conjecture of Wormald ([45], Conjecture 2.27) that minimum degree 3 suffices. Two other natural questions of great interest to me are on the chromatic number and independence number of Gz . Problem 3 For a given z, what is the chromatic number of Gz ? Problem 4 For a given z, what is the independence number of Gz ? For random regular graphs, Problem 3 is almost completely resolved (see [39] and [1]). For Gz , substantial progress has been made as well [16]. Problem 4, on the other hand, remains open even for z = (0, 0, 1), i.e. random 3-regular graphs, with the best current bounds of .4352 ≤ α(G(0,0,1) )/n w.h.p. and for G(0,0,1) of sufficiently large girth, α(G(0,0,1) )/n ≤ .4554 (see Kardoˇs, Kr´al and Volec [22] for the lower bound and McKay [26] for the upper bound). I am working on a refinement of McKay’s method that appears promising in reducing the .4554 upper bound on α(G(0,0,1) )/n for graphs of sufficiently large girth, although it does not appear to close the gap completely.

2

The H-free Process

The differential equations method has become an invaluable tool in the study of Gz (see [5], [28], or [39], for example), as well as in the analysis of other random graph processes. The idea of the method is fairly simple: given a collection of random variables, defined with respect to a random process, treat each variable as a continuous function and the expected one-step changes in each as its derivative. If these expected changes can be described

2

as continuous functions in these variables, one obtains a system of ordinary differential equations. Large deviation inequalities can then be applied to show these approximations are ‘correct’. I direct the reader to Wormald’s monograph [44] for details as well as several applications. One application of this method I find particularly fascinating is the recent analysis of the triangle-free process by Bohman [4], and the subsequent analysis of the H-free process by Bohman and Keevash [6]. For a fixed graph H, the H-free process is as follows. Beginning with a graph G0 on n isolated vertices, construct a sequence of graphs G0 , G1 , . . . as follows: Gi+1 is constructed by adding an edge ei+1 to Gi , where ei+1 is chosen uniformly at random from all pairs of vertices which neither form edges of Gi nor create a (not-necessarily induced) copy of H if added to Gi . This process terminates with a maximal H-free graph GH with a random number MH of edges. The H-free process was introduced by Erd˝os, Suen and Winkler [13], who established bounds on the size and and independence number of the triangle-free process that led to the best lower bounds on the Ramsey number R(3, t) known at the time. However, the problem of determining the correct asymptotic magnitude of MK3 remained open until Bohman (via the differential equations method) established the following. p Theorem 1 (Bohman [4]) W.h.p., MK3 = Θ( log(n) · n3/2 ). Theorem 2 (Bohman [4]) There is a constant c such that the following holds: if n = n(t) < c ·

t2 , log(t)

then w.h.p. the triangle-free process on n vertices produces a graph with no independent set of cardinality t, for t sufficiently large. Theorem 1 is the first result producing the asymptotic final size of an H-free process for any graph H containing a cycle. Theorem 2 provides a new proof of the celebrated result of Kim [25], that the Ramsey number R(3, t) ≥ C · t2 / log(t) for some C > 0, with t sufficiently large. A graph H is said to be 2-balanced if (e(H) − 1)/(v(H) − 2) ≥ (e(F ) − 1)/(v(F ) − 2) for all proper subgraphs F of H with at least 3 vertices, and strictly 2-balanced if the inequality is sharp for all such F . Examples of strictly 2-balanced graphs include cycles and complete graphs. Extending the methods of [4], Bohman and Keevash [6] established new lower bounds on MH for strictly 2-balanced H, as well as upper bounds on the independence number of GH when H is a cycle or complete graph, all of which hold with high probability, yielding new asymptotic lower bounds for a collection of Ramsey and Tur´an numbers. For H containing a cycle, the best likely upper bounds on MH to date remain a logarithmic factor larger than the lower bounds given in [6] (see [31]) - thus, determining the correct asymptotic order of MH remains an important open problem. For the complete graph K4 , this has recently been resolved by Warnke [42] and Wolfovitz [43], independently, who produced upper bounds matching Bohman’s lower bound [4] to within a constant factor. In analyzing the C` -free process, where C` denotes the cycle of length `, I have been able to determine both the correct asymptotic final size and independence number. Theorem 3 (P. [34],[35]) Let ` ≥ 4 be a fixed integer. With high probability, MC` = Θ(n`/(`−1) (log n)1/`−1 ) and α(GC` ) = Θ((n log n)1/(`−1) ). 3

This result was also independently found by Warnke [41]. However, the analysis in [6] only extends to graphs H which are strictly 2-balanced; for graphs which are 2-balanced but not strictly so, much less is known. Among such graphs containing a cycle, the smallest example is the diamond graph K4− , formed by removing an edge from K4 . Through a careful modification of the techniques employed in [4] and [6], I have been able to determine the asymptotic final size and independence number of the diamond-free process. p Theorem 4 (P. [32]) With high probability, MK − = Θ( log(n) · n3/2 ) and α(GK − ) = 4

4

Θ((n log n)1/2 ). I am actively working on extending these techniques to other choices of H, as well as on applying related techniques to analyze similar random processes.

3

Extremal Set Theory

Extremal set theory, on the whole, is the study of the maximum (or minimum) size of families of sets satisfying certain conditions. Perhaps the most famous and fundamental result in extremal set theory is the Erd˝os-Ko-Rado Theorem (See 8.1 in [20]): Theorem 5 (Erd˝ os-Ko-Rado [12]) If n and k are positive integers with n ≥ 2k, and F is a family
of k-element subsets of an n-element set with
A ∩ B 6= ∅ for all A, B ∈ F, then n−1 |F| ≤ n−1 . Furthermore, if n > 2k and |F| = k−1 k−1 , then F consists of all k-element subsets containing a given element. Various generalizations of the Erd˝os-Ko-Rado Theorem have been proposed, including the following by Chv´ atal [9]. Define a d-simplex to be a collection of d + 1Tdistinct sets, A1 , . . . , Ad+1 , such that any d of them have nonempty intersection, but d+1 i=1 Ai = ∅. Additionally, let [n] = {1, 2, . . . , n}. Let f (n, k, d) be the maximum size of a family F of k-element subsets of [n] containing no d-simplex. Thus, by Theorem 5, if n ≥ 2k,
f (n, k, 1) = n−1 . k−1
Erd˝os [11] conjectured that f (n, k, 2) = n−1 k−1 , which was recently settled by Mubayi and Verstra¨ete [29]. Chv´ atal, in attacking Erd˝os’s conjecture, proved the following:
Theorem 6 (Chv´ atal [9]) If n ≥ k + 2 ≥ 5, then f (n, k, k − 1) = n−1 k−1 . In the same paper, he posed the following conjecture.
n−1 Conjecture 2 (Chv´ atal [9]) If k ≥ d + 1 ≥ 2 and n > ( d+1 d )k, then f (n, k, d) = k−1 . Moreover, if F is a family of k-element subsets of [n] containing no d-simplex and |F| =
n−1 , then F = {A ⊂ [n] : |A| = k, a ∈ A} for some a ∈ [n]. k−1 Progress on Conjecture 2 was made by Frankl and F¨ uredi [14], who confirmed it when n is a sufficiently large function of k, and by Keevash and Mubayi [23], who confirmed it for sufficiently large n in the case that k and n/2 − k are both bounded away from 0. If one removes the restriction that every member of the family has the same cardinality, significantly more is known. As a 1-simplex is a pair of nonempty disjoint sets, the bound becomes 2n−1 + 1, with many extremal families. The question of forbidding 2-simplices was 4

resolved by Milner (unpublished, see the references in [29]); it is known that the maximumsize family consists of all subsets either
containing a fixed element or of cardinality at most
n−1 2, so the resulting bound is 2n−1 + n−1 + . (The characterization of the extremal fam0 1 ily first explicitly appeared in [29].) Keevash and Mubayi [23] generalized this to that Pshow d−1 n−1
n−1 for any fixed d ≥ 3 and n = n(d) sufficiently large, the correct bound is 2 + i=0 i , satisfied only by families consisting of all subsets either containing a given element or of size at most d − 1. They further conjectured that for any d ≥ 3, this bound and extremal family is correct for all n. I have been able to verify this conjecture when d = 3 [37]. Another problem of interest to me, related to the above question, was posed by Erd˝ os [11]. For a family F of subsets of [n], write F → K3 if there are three sets A1 , A2 , A3 ∈ F and three elements a1 , a2 , a3 ∈ [n] such that Ai ∩ {a1 , a2 , a3 } = {a1 , a2 , a3 } \ {ai }. Equivalently, F → K3 if the family {A ∩ {a1 , a2 , a3 } | A ∈ F} contains the complete graph K3 on {a1 , a2 , a3 }, for some choice of a1 , a2 , a3 . Let g(n) denote the maximum integer m such that there exists a family F of subsets of [n] with |F| = m and F 6→ K3 . The problem posed
was to determine g(n) for all n. Anstee [2],[3] resolved this by showing g(n) = 1 + n + n2 through linear algebraic techniques, and, furthermore, that there are exponentially many extremal families. Consider now a further restriction: let g(n, k) denote the same function with the additional condition that F consists only of k-element subsets of [n]. By Tur´an’s Theorem, one can see that g(n, 2) = bn2 /4c. A result of Frankl and Pach [15] yields that this bound generalizes: Theorem 7 (Frankl and Pach [15]) For n ≥ k ≥ 3,   (n − k + 2)2 g(n, k) = . 4 Furthermore, if F is extremal, i.e. F 6→ K3 and |F| = g(n, k), there is a k − 2 element subset X such that X ⊂ A for all A ∈ F, and {A \ X | A ∈ F} ∼ = Kb(n−k+2)/2c,d(n−k+2)/2e , the maximum-size complete bipartite graph on n − k + 2 elements. In analyzing this problem, I was able to provide a new proof of Theorem 7, as well as the following generalization for sufficiently large n: Theorem 8 (P. [33]) If n ≥ n0 (k), and F is a family of k-element subsets of [n] with F 6→ K3 and (n − k + 1)2 , |F| > 4 then there is a k − 2 element subset X such that X ⊂ A for all A ∈ F, and this bound is best possible. Moreover, the family {A \ X | A ∈ F} can be made bipartite by removing at most one set from F.

5

I am working with Oleg Pikhurko on an extension of Theorem 8. The underlying analysis relates the above problem to the question of determining the minimum integer m(n, i) such that any triangle-free graph on n vertices with m(n, i) edges can be made bipartite by deleting at most i vertices (or, equivalently, contains an induced bipartite subgraph on at least n − i vertices). We have successfully determined m(n, i) for all n ≥ 5, i < bn/5c, and are currently working on establishing bounds for i ≥ bn/5c. The arguments used in the above analysis also led to a new and simpler proof of a result of Hanson and Toft [17] classifying the maximum-size Kr -free graphs with chromatic number at least r [36].

References [1] D. Achlioptas and C. Moore, The chromatic number of random regular graphs, in Proc. 8th Int. Workshop on Randomization and Approximation Techniques in Computer Science (2004), pp. 219-228. [2] R. Anstee, Properties of (0,1)-matrices with no triangles, J. Combin. Theory Ser. A 29 (1980), 186-198. [3] R. Anstee, Hypergraphs with no special cycles, Combinatorica 3 (1983), 141-146. [4] T. Bohman, The triangle-free process, Adv. Math. 221 (2009), 1653-1677. [5] T. Bohman and A. Frieze, Karp-Sipser on random graphs with a fixed degree sequence, Combin. Probab. Comput. 20 (2011), 721-741. [6] T. Bohman and P. Keevash, The early evolution of the H-free process, Invent. Math. 181 (2010), 291-336. [7] T. Bohman and M. Picollelli, SIR epidemics on random graphs with a fixed degree sequence, Random Struct. Alg., to appear. (Available online at http://sites.google.com/site/mepicollelli ) [8] B. Bollob´ as, Random Graphs, Cambridge University Press, 2001. [9] V. Chv´ atal, An extremal set-intersection theorem, J. London Math. Soc. 9 (1974), 355-359. [10] F. Chung, L. Lu and V. Vu, The spectra of random graphs with given expected degrees, Proceedings of National Academy of Sciences 100 (2003), 6313-6318. [11] P. Erd˝ os, Topics in combinatorial analysis, in Proc. Second Louisiana Conf. on Comb., Graph Theory and Computing, R. C. Mullin et al. (eds.), pp. 2-20. Louisiana State University, Baton Rouge, 1971. [12] P. Erd˝ os, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. (2) 12 (1961), 313-320. [13] P. Erd˝ os, S. Suen and P. Winkler, On the size of a random maximal graph, Random Struct. Alg. 6 (1995), 309-318. [14] P. Frankl and Z. F¨ uredi, Exact solution of some Tur´an-type problems, J. Combin. Theory Ser. A. 45 (1987), 226-262. [15] P. Frankl and J. Pach, On disjointly representable sets, Combinatorica 4 (1984), 39-45. [16] A. Frieze, M. Krivelevich and C. Smyth, On the chromatic number of random graphs with a fixed degree sequence, Combin. Probab. Comput. 16 (2007), 733-746. [17] D. Hanson and B. Toft, k-saturated graphs of chromatic number at least k, Ars Combinatoria 31 (1991), 159-164. [18] S. Janson, The largest component in a subcritical random graph with a power law degree distribution, Ann. Appl. Probab. 18 (2008), 1651-1668. [19] S. Janson, T. Luczak and A. Ruci´ nski, Random Graphs, John Wiley and Sons, 2000. [20] S. Jukna, Extremal Combinatorics With Applications in Computer Science, Springer-Verlag, 2001.

6

[21] M. Kang and T. Seierstad, The critical phase for random graphs with a given degree sequence, Combin. Probab. Comput. 17 (2008), 67-86. [22] F. Kardoˇs, D. Kr´ al, and J. Volec, Fractional colorings of cubic graphs with large girth, manuscript. arXiv:1010.3415 [23] P. Keevash and D. Mubayi, Set systems without a simplex or a cluster, Combinatorica 30 (2010), 175-200. [24] W. O. Kermack and A. G. McKendrick, A Contribution to the Mathematical Theory of Epidemics, Proc. Roy. Soc. Lond. A 115 (1927), 700-721. [25] J. H. Kim, The Ramsey number R(3, t) has order of magnitude t2 / log(t), Random Struct. Alg. 7, (1995), 173-207. [26] B. D. McKay, Independent sets in regular graphs of high girth, Ars Combinatoria 23A (1987), 179-185. [27] M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, Random Struct. Alg. 6 (1996), 161-180. [28] M. Molloy and B. Reed, The size of the largest component of a random graph on a fixed degree sequence, Combin. Probab. Comput. 7 (1998), 295-306. [29] D. Mubayi and J. Verstra¨ete, Proof of a conjecture of Erd˝os on triangles in set-systems, Combinatorica 25 (2005), 599-614. [30] M. Newman, Random graphs as models of networks, in Handbook of Graphs and Networks, S. Bornholdt and H. G. Schuster (eds.), pp. 35-65. Wiley-VCH, Berlin, 2002. [31] D. Osthus and A. Taraz, Random maximal H-free graphs, Random Struct. Alg. 18 (2001), 61-82. [32] M. Picollelli, The diamond-free process, Random Struct. Alg., to appear. (Preprint available at http://sites.google.com/site/mepicollelli ) [33] M. Picollelli, Extremal problems and random processes on graphs, PhD thesis, Carnegie Mellon University, (2008). [34] M. Picollelli, The final size of the C4 -free process, Combin. Probab. Comput. 20 (2011), 939-955. [35] M. Picollelli, The final size of the C` -free process, submitted. (Available online at http://sites.google.com/site/mepicollelli ) [36] M. Picollelli, A note on Kr -free graphs with chromatic number at least r, submitted. (Available online at http://sites.google.com/site/mepicollelli ) [37] M. Picollelli, Set systems without a 3-simplex, Discrete Math. 311 (2011), 2113-2116. [38] B. Pittel, The largest component in a subcritical random graph with a power law degree distribution, Ann. Appl. Probab. 18 (2008), 1636-1650. [39] L. Shi and N. Wormald, Colouring random regular graphs, Combin. Probab. Comput. 16 (2007), 459-494. [40] E. Volz, SIR dynamics in random networks with heterogeneous connectivity, J. Math. Biol. 56 (2007), 293-310. [41] L. Warnke, The C` -free process, manuscript. arXiv:1101.0693 [42] L. Warnke, When does the K4 -free process stop?, manuscript. arXiv:1007.3037 [43] G. Wolfovitz, The K4 -free process, manuscript. arXiv:1008.4044 [44] N.C. Wormald, The differential equation method for random graph processes and greedy algorithms, in Lectures on Approximation and Randomized Algorithms, M. Karo´ nski and H.J. Pr¨ omel (eds.), pp. 73-155. PWN, Warsaw, 1999. [45] N.C. Wormald, Models of random regular graphs, in Surveys in Combinatorics, 1999 (Canterbury), J.D. Lamb and D.A. Preece (eds.), pp. 239-298. Cambridge University Press, Cambridge, 1999. [46] R.W. Robinson and N.C. Wormald, Almost all regular graphs are hamiltonian, Random Struct. Alg. 5 (1994), 363-374.

7