Publications (47)17.93 Total impact
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ABSTRACT: Answering a question of Kolaitis and Kopparty, we show that, for given integer $q>1$ and pairwise nonisomorphic connected graphs $G_1\dots G_k$, if $p=p(n) $ is such that $\Pr(G_{n,p}\supseteq G_i)\to \infty$ $\forall i$, then, with $\xi_i$ the number of copies of $G_i$ in $G_{n,p}$, $(\xi_1\dots \xi_k)$ is asymptotically uniformly distributed on $\mathbf{Z}_q^k$.02/2014;  [show abstract] [hide abstract]
ABSTRACT: Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph $G=G_{n,p}$, we show, roughly speaking, that (with high probability) the triangles of $G$ span its cycle space whenever each of its edges lies in a triangle (which happens (w.h.p.) when $p$ is at least about $\sqrt{(3/2)\ln n/n}$, and not below this unless $p$ is very small.) We give two related proofs of this statement, together with a relatively simple proof of a fundamental "stability" theorem for trianglefree subgraphs of $G_{n,p}$, originally due to Kohayakawa, \L uczak and R\"odl, that underlies the first of our proofs.07/2012; 
Article: Mantel's Theorem for random graphs
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ABSTRACT: For a graph $G$, denote by $t(G)$ (resp. $b(G)$) the maximum size of a trianglefree (resp. bipartite) subgraph of $G$. Of course $t(G) \geq b(G)$ for any $G$, and a classic result of Mantel from 1907 (the first case of Tur\'an's Theorem) says that equality holds for complete graphs. A natural question, first considered by Babai, Simonovits and Spencer about 20 years ago is, when (i.e. for what $p=p(n)$) is the "Erd\H{o}sR\'enyi" random graph $G=G(n,p)$ likely to satisfy $t(G) = b(G)$? We show that this is true if $p>C n^{1/2} \log^{1/2}n $ for a suitable constant $C$, which is best possible up to the value of $C$.Random Structures and Algorithms 06/2012; · 1.05 Impact Factor  [show abstract] [hide abstract]
ABSTRACT: We give upper bounds for the number $\Phi_\ell(G)$ of matchings of size $\ell$ in (i) bipartite graphs $G=(X\cup Y, E)$ with specified degrees $d_x$ ($x\in X$), and (ii) general graphs $G=(V,E)$ with all degrees specified. In particular, for $d$regular, $N$vertex graphs, our bound is best possible up to an error factor of the form $\exp[o_d(1)N]$, where $o_d(1) \rightarrow 0$ as $d \rightarrow \infty$. This represents the best progress to date on the "Upper Matching Conjecture" of Friedland, Krop, Lundow and Markstr\"om. Some further possibilities are also suggested.Journal of Combinatorial Theory, Series A. 05/2012; 120(5).  [show abstract] [hide abstract]
ABSTRACT: Answering several questions of Duffus, Frankl and R\"odl, we give asymptotics for the logarithms of (i) the number of maximal antichains in the ndimensional Boolean algebra and (ii) the numbers of maximal independent sets in the covering graph of the ndimensional hypercube and certain natural subgraphs thereof. The results in (ii) are implied by more general upper bounds on the numbers of maximal independent sets in regular and biregular graphs. We also mention some stronger possibilities involving actual rather than logarithmic asymptotics.Order 02/2012; 30(2). · 0.40 Impact Factor 
Article: Upper Tails for Cliques
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ABSTRACT: With $\xi_{k}=\xi_{k}^{n,p}$ the number of copies of $K_k$ in the usual (Erd\H{o}sR\'enyi) random graph $G(n,p)$, $p\geq n^{2/(k1)}$ and $\eta>0$, we show when $k>1$ $$\Pr(\xi_k> (1+\eta)\E \xi_k) < \exp [\gO_{\eta,k} \min\{n^2p^{k1}\log(1/p), n^kp^{\binom{k}{2}}\}].$$ This is tight up to the value of the constant in the exponent.11/2011;  [show abstract] [hide abstract]
ABSTRACT: Let be a set of terms over an arbitrary (but finite) number of Boolean variables. Let U() be the set of truth assignments that satisfy exactly one term in . Motivated by questions in computational complexity, Rudich conjectured that there exist ∊, δ > 0 such that, if is any set of terms for which U() contains at least a (1−∊)fraction of all truth assignments, then there exists a term t ∈ such that at least a δfraction of assignments satisfy some term of sharing a variable with t [8].We prove a stronger version: for any independent assignment of the variables (not necessarily the uniform one), if the measure of U() is at least 1 − ∊, there exists a t ∈ such that the measure of the set of assignments satisfying either t or some term incompatible with t (i.e., having no satisfying assignments in common with t) is at least . (A key part of the proof is a correlationlike inequality on events in a finite product probability space that is in some sense dual to Reimer's inequality [11], a.k.a. the BKR inequality [5], or the van den Berg–Kesten conjecture [3].)Combinatorics Probability and Computing 01/2011; 20:257266. · 0.61 Impact Factor 
Article: Upper tails for triangles
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ABSTRACT: With $\xi$ the number of triangles in the usual (Erd\H{o}sR\'enyi) random graph $G(m,p)$, $p>1/m$ and $\eta>0$, we show (for some $C_{\eta}>0$) $$\Pr(\xi> (1+\eta)\E \xi) < \exp[C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}].$$ This is tight up to the value of $C_{\eta}$.05/2010; 
Article: The number of 3SAT functions
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ABSTRACT: With $G_k(n)$ the number of functions of $n$ boolean variables definable by $k$SAT formulae, we prove that $G_3(n)$ is asymptotic to $2^{n+\binom{n}{3}}$. This is a strong form of the case $k=3$ of a conjecture of Bollob\'as, Brightwell and Leader stating that for fixed $k$, $\log_2 G_k(n)\sim \binom{n}{k}$. Comment: 51 pagesIsrael Journal of Mathematics 05/2010; · 0.65 Impact Factor  [show abstract] [hide abstract]
ABSTRACT: Competing urns refers to the random experiment where m balls are dropped, randomly and independently, into urns 1,...,n. Formally, we have a random map $\sigma$ from {1,...,m} to {1,...,n} with the $\sigma(i)$'s i.i.d. With $x_j$ the indicator of the event that at least $t_j$ balls land in urn j (for some threshold $t_j$), we prove conditional negative association for the random variables $x_1,...,x_n$. We mostly deal with the more general situation in which the $\sigma(i)$'s need not be identically distributed, proving results which imply conditional negative association in the i.i.d. case. Some of the resultsparticularly Lemma 8 on graph orientationsare thought to be of independent interest. We also give a counterexample to a negative correlation conjecture of D. Welsh, a strong version of a (still open) conjecture of G. Farr. Comment: 19 pagesRandom Structures and Algorithms 01/2010; · 1.05 Impact Factor 
Article: On the Number of 2SAT Functions
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ABSTRACT: We give an alternative proof of a conjecture of Bollobás, Brightwell and Leader, first proved by Peter Allen, stating that the number of Boolean functions definable by 2SAT formulae is . One step in the proof determines the asymptotics of the number of ‘oddbluetrianglefree’ graphs on n vertices.Combinatorics Probability and Computing 08/2009; 18(05):749  764. · 0.61 Impact Factor  [show abstract] [hide abstract]
ABSTRACT: We introduce the antipodal pairs property for probability measures on finite Boolean algebras and prove that conditional versions imply strong forms of logconcavity. We give several applications of this fact, including improvements of some results of Wagner; a new proof of a theorem of Liggett stating that ultralogconcavity of sequences is preserved by convolutions; and some progress on a wellknown logconcavity conjecture of J. Mason. Comment: 13 pages07/2009; 
Article: Hamiltonian cycles in Dirac graphs.
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ABSTRACT: We prove that for any nvertex Dirac graph (graph with minimum degree at least n/2) G=(V,E), the number, Ψ(G), of Hamiltonian cycles in G is at least $$ exp_2 [2h(G)  n\log e  o(n)], $$ where h(G)=maxΣ e x e log(1/x e ), the maximum over x: E → ℜ+ satisfying Σ e∋υ x e = 1 for each υ ∈ V, and log =log2. (A second paper will show that this bound is tight up to the o(n).) We also show that for any (Dirac) G of minimum degree at least d, h(G) ≥ (n/2) logd, so that Ψ(G) > (d/(e + o(1))) n . In particular, this says that for any Dirac G we have Ψ(G) > n!/(2 + o(1)) n , confirming a conjecture of G. Sárközy, Selkow, and Szemerédi which was the original motivation for this work.Combinatorica 01/2009; 29:299326. · 0.56 Impact Factor  [show abstract] [hide abstract]
ABSTRACT: For a graph G = (V,E) and x: E → ℜ+ satisfying Σ e∋υ x e = 1 for each υ ∈ V, set h(x) = Σ e x e log(1/x e ) (with log = log2). We show that for any nvertex G, random (not necessarily uniform) perfect matching f satisfying a mild technical condition, and x e =Pr(e∈f), $$ H(f) < h(x)  \frac{n} {2}\log e + o(n) $$ (where H is binary entropy). This implies a similar bound for random Hamiltonian cycles. Specializing these bounds completes a proof, begun in [6], of a quite precise determination of the numbers of perfect matchings and Hamiltonian cycles in Dirac graphs (graphs with minimum degree at least n/2) in terms of h(G):=maxΣ e x e log(1/x e ) (the maximum over x as above). For instance, for the number, Ψ(G), of Hamiltonian cycles in such a G, we have $$ \Psi (G) = exp_2 [2h(G)  n\log e  o(n)]. $$ .Combinatorica 01/2009; 29:327335. · 0.56 Impact Factor 
Article: Factors in random graphs.
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ABSTRACT: Let $H$ be a fixed graph on $v$ vertices. For an $n$vertex graph $G$ with $n$ divisible by $v$, an $H${\em factor} of $G$ is a collection of $n/v$ copies of $H$ whose vertex sets partition $V(G)$. In this paper we consider the threshold $th_{H} (n)$ of the property that an Erd\H{o}sR\'enyi random graph (on $n$ points) contains an $H$factor. Our results determine $th_{H} (n)$ for all strictly balanced $H$. The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random $k$uniform hypergraph, solving the wellknown "Shamir's problem."Random Struct. Algorithms. 01/2008; 33:128.  [show abstract] [hide abstract]
ABSTRACT: We give counterexamples and a few positive results related to several conjectures of R. Pemantle and D. Wagner concerning negative correlation and logconcavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. We also give short proofs of a pair of results due to Pemantle and Borcea et al.; prove that "almost exchangeable" measures satisfy the "FederMihail" property, thus providing a "nonobvious" example of a class of measures for which this important property can be shown to hold; and mention some further questions. Comment: 21 pages; only minor changes since previous version; accepted for publication in Random Structures and AlgorithmsRandom Structures and Algorithms 12/2007; · 1.05 Impact Factor  [show abstract] [hide abstract]
ABSTRACT: A fractional fuzzy Potts measure is a probability distribution on spin configurations of a finite graph $G$ obtained in two steps: first a subgraph of $G$ is chosen according to a random cluster measure $\phi_{p,q}$, and then a spin ($\pm1$) is chosen independently for each component of the subgraph and assigned to all vertices of that component. We show that whenever $q\geq1$, such a measure is positively associated, meaning that any two increasing events are positively correlated. This generalizes earlier results of H\"{a}ggstr\"{o}m [Ann. Appl. Probab. 9 (1999) 11491159] and H\"{a}ggstr\"{o}m and Schramm [Stochastic Process. Appl. 96 (2001) 213242].The Annals of Probability 11/2007; 35(6). · 1.38 Impact Factor  [show abstract] [hide abstract]
ABSTRACT: Consider ordinary bond percolation on a finite or countably infinite graph. Let s, t, a, and b be vertices. An earlier paper (J. Van den Berg and J. Kahn, Ann Probab 29 (2001), 123–126) proved the (nonintuitive) result that, conditioned on the event that there is no open path from s to t, the two events ”there is an open path from s to a” and “there is an open path from s to b” are positively correlated. In the present paper we further investigate and generalize the theorem of which this result was a consequence. This leads to results saying, informally, that, with the above conditioning, the open cluster of s is conditionally positively (self) associated and that it is conditionally negatively correlated with the open cluster of t.We also present analogues of some of our results for (a) randomcluster measures and (b) directed percolation and contact processes and observe that the latter lead to improvements of some of the results in a paper of Belitsky et al. (Stoch Proc Appl 67 (1997), 213–225). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006Random Structures and Algorithms 11/2006; 29(4):417  435. · 1.05 Impact Factor  [show abstract] [hide abstract]
ABSTRACT: Consider the onedimensional contact process. About ten years ago, N. Konno stated the conjecture that, for all positive integers $n,m$, the upper invariant measure has the following property: Conditioned on the event that $O$ is infected, the events $\{$All sites $n,...,1$ are healthy$\}$ and $\{$All sites $1,...,m$ are healthy$\}$ are negatively correlated. We prove (a stronger version of) this conjecture, and explain that in some sense it is a dual version of the planar case of one of our results in \citeBHK.09/2006;  [show abstract] [hide abstract]
ABSTRACT: We consider relations between thresholds for monotone set properties and simple lower bounds for such thresholds. A motivating example (Conjecture 2): Given an nvertex graph H, write pE for the least p such that, for each subgraph H' of H, the expected number of copies of H' in G=G(n, p) is at least 1, and pc for that p for which the probability that G contains (a copy of) H is 1/2. Then (conjecture) pc=O(pElog n). Possible connections with discrete isoperimetry are also discussed.Combinatorics Probability and Computing 04/2006; · 0.61 Impact Factor
Publication Stats
851  Citations  
17.93  Total Impact Points  
Top Journals
Institutions

1983–2012

Rutgers, The State University of New Jersey
 Department of Mathematics
New Brunswick, New Jersey, United States


1997

University of Montana
 Department of Mathematical Sciences
Missoula, MT, United States


1988–1994

Hebrew University of Jerusalem
 • Einstein Institute of Mathematics
 • Rachel and Selim Benin School of Computer Science and Engineering
Jerusalem, Jerusalem District, Israel


1986

Hungarian Academy of Sciences
Budapeŝto, Budapest, Hungary


1984

Massachusetts Institute of Technology
Cambridge, Massachusetts, United States
