In this paper, we consider a randomized greedy algorithm for independent sets in r‐uniform d‐regular hypergraphs G on n vertices with girth g. By analyzing the expected size of the independent sets generated by this algorithm, we show that , where converges to 0 as g → ∞ for fixed d and r, and f(d, r) is determined by a differential equation. This extends earlier results of Garmarnik and Goldberg for

We study the probabilistic degree over of the OR function on n variables. For , the ‐error probabilistic degree of any Boolean function f : {0, 1}n → {0, 1} over is the smallest nonnegative integer d such that the following holds: there exists a distribution P of polynomials of degree at most d such that for all , we have . It is known from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel

We show that throughout the satisfiable phase the normalized number of satisfying assignments of a random 2‐SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. The proof is based on showing that the Belief Propagation algorithm renders the correct marginal probability that a variable is set to “true” under a uniformly random satisfying assignment

This is concerned with voting processes on graphs where each vertex holds one of two different opinions. In particular, we study the Best‐of‐two and the Best‐of‐three. Here at each synchronous round, each vertex updates its opinion to match the majority among the opinions of two random neighbors and itself (the Best‐of‐two) or the opinions of three random neighbors (the Best‐of‐three). In this study

The r‐size‐Ramsey number of a graph H is the smallest number of edges a graph G can have such that for every edge‐coloring of G with r colors there exists a monochromatic copy of H in G. For a graph H, we denote by Hq the graph obtained from H by subdividing its edges with q − 1 vertices each. In a recent paper of Kohayakawa, Retter and Rödl, it is shown that for all constant integers q, r ≥ 2 and

The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an Erdős‐Rényi random graph exceeds its expectation by some constant factor. There has been much exciting recent progress on this problem. We study the corresponding problem for hypergraphs, for which less is known about the large deviation rate. We present new phenomena in

The triangle‐free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal triangle‐free graph at which the triangle‐free process terminates. We also bound the independence number of this graph, which gives an improved lower bound on the Ramsey

The Kohayakawa–Nagle–Rödl‐Schacht conjecture roughly states that every sufficiently large locally d‐dense graph G on n vertices must contain at least (1 − o(1))d|E(H )|n|V (H )| copies of a fixed graph H. Despite its important connections to both quasirandomness and Ramsey theory, there are very few examples known to satisfy the conjecture. We provide various new classes of graphs that satisfy the

We describe a new random greedy algorithm for generating regular graphs of high girth: Let k ≥ 3 and c ∈ (0, 1) be fixed. Let ℕ be even and set . Begin with a Hamilton cycle G on n vertices. As long as the smallest degree , choose, uniformly at random, two vertices u, v ∈ V(G) of degree whose distance is at least g − 1. If there are no such vertex pairs, abort. Otherwise, add the edge uv to E(G). We

Let X1, … , Xn be independent identically distributed discrete random vectors in . We consider upper bounds on under various restrictions on Xi and weights ai. When , this corresponds to the classical Littlewood‐Offord problem. We prove that in general for identically distributed random vectors and even values of n the optimal choice for (ai) is ai = 1 for and ai = −1 for , regardless of the distribution

We consider 2‐dimensional random simplicial complexes Y in the multiparameter model. We establish the multiparameter threshold for the property that every 2‐dimensional simplicial complex S admits a topological embedding into Y asymptotically almost surely. Namely, if in the procedure of the multiparameter model on n vertices, each i‐dimensional simplex is taken with probability pi = pi(n), then the

The beta polytope is the convex hull of n i.i.d. random points distributed in the unit ball of according to a density proportional to if (in particular, corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if . We show that the expected normalized volumes of high‐dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results

Bollobás‐Riordan random pairing model of a preferential attachment graph is studied. Let {Wj}j ≤ mn + 1 be the process of sums of independent exponentials with mean 1. We prove that the degrees of the first vertices are jointly, and uniformly, asymptotic to , and that with high probability (whp) the smallest of these degrees is , at least. Next we bound the probability that there exists a pair of large

We consider permutations avoiding a pattern of length three under the family of Mallows distributions. In particular, for any pattern , we obtain rather precise results on the asymptotic probability as n → ∞ that a permutation under the Mallows distribution with parameter q ∈ (0, 1) avoids the pattern. By a duality between the parameters q and , we also obtain rather precise results on the above probability

We study random permutations of n objects with respect to multiplicative measures with polynomial growing cycle weights. We determine in this paper the asymptotic behavior of the long cycles under this measure and also prove that the cumulative cycle numbers converge in the region of the long cycles to a Poisson process.

Let fs, k(n) be the maximum possible number of s‐term arithmetic progressions in a set of n integers which contains no k‐term arithmetic progression. For all fixed integers k > s ≥ 3, we prove that fs, k(n) = n2 − o(1), which answers an old question of Erdős. In fact, we prove upper and lower bounds for fs, k(n) which show that its growth is closely related to the bounds in Szemerédi's theorem.

Property testers are fast randomized algorithms whose task is to distinguish between inputs satisfying some predetermined property and those that are far from satisfying it. A landmark result of Alon et al. states that for any finite family of graphs , the property of being induced ‐free (i.e., not containing an induced copy of any ) is testable. Goldreich and Shinkar conjectured that one can extend

A perfect Kr‐tiling in a graph G is a collection of vertex‐disjoint copies of Kr that together cover all the vertices in G. In this paper we consider perfect Kr‐tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze, and Martin [7] where one starts with a dense graph and then adds m random edges to it. Specifically, given any fixed we determine how many random edges

In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product , where is a real centered uniformly bounded random variable of unit variance and bn is an independent Bernoulli random variable with a probability

We give a detailed asymptotic analysis of the profiles of random symmetric digital search trees, which are in close connection with the performance of the search complexity of random queries in such trees. While the expected profiles have been analyzed for several decades, the analysis of the variance turns out to be very difficult and challenging, and requires the combination of several different

In a seminal paper on finding large matchings in sparse random graphs, Karp and Sipser proposed two algorithms for this task. The second algorithm has been intensely studied, but due to technical difficulties, the first algorithm has received less attention. Empirical results by Karp and Sipser suggest that the first algorithm is superior. In this paper we show that this is indeed the case, at least

We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for

We study the fixation time of the identity of the leader, that is, the most massive component, in the general setting of Aldous's multiplicative coalescent, which in an asymptotic sense describes the evolution of the component sizes of a wide array of near‐critical coalescent processes, including the classical Erdős‐Rényi process. We show tightness of the fixation time in the “Brownian” regime, explicitly

We consider column‐sparse covering integer programs, a generalization of set cover. We develop a new rounding scheme based on the partial resampling variant of the Lovász Local Lemma developed by Harris and Srinivasan. This achieves an approximation ratio of , where amin is the minimum covering constraint and is the maximum ℓ1‐norm of any column of the covering matrix A (whose entries are scaled to

We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables. This result is a first step for giving a nonparametric test for identifying the degree function of a large random graph. Replacing the indicator function in the empirical CDF by a smoother function

In this paper we consider the existence of Hamilton cycles in the random graph . This random graph is chosen uniformly from , the set of graphs with vertex set [n], m edges and minimum degree at least 3. Our ultimate goal is to prove that if m = cn and c > 3/2 is constant then G is Hamiltonian w.h.p. In Frieze (2014), the second author showed that c ≥ 10 is sufficient for this and in this paper we

We consider an online vector balancing game where vectors vt, chosen uniformly at random in {− 1, + 1}n, arrive over time and a sign xt ∈ {− 1, + 1} must be picked immediately upon the arrival of vt. The goal is to minimize the L∞ norm of the signed sum . We give an online strategy for picking the signs xt that has value O(n1/2) with high probability. Up to constants, this is the best possible even

In this paper, we study the following recently proposed semi‐random graph process: starting with an empty graph on n vertices, the process proceeds in rounds, where in each round we are given a uniformly random vertex v, and must immediately (in an online manner) add to our graph an edge incident with v. The end goal is to make the constructed graph satisfy some predetermined monotone graph property

It is known that a sequence of permutations is quasirandom if and only if the pattern density of every 4‐point permutation in converges to 1/24. We show that there is a set S of 4‐point permutations such that the sum of the pattern densities of the permutations from S in the permutations converges to if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets

A well‐known result of Rödl and Ruciński states that for any graph H there exists a constant C such that if , then the random graph Gn, p is a.a.s. H‐Ramsey, that is, any 2‐coloring of its edges contains a monochromatic copy of H. Aside from a few simple exceptions, the corresponding 0‐statement also holds, that is, there exists c > 0 such that whenever the random graph Gn, p is a.a.s. not H‐Ramsey

We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short.

Given graphs F, H and G, we say that G is (F, H )v‐Ramsey if every red/blue vertex coloring of G contains a red copy of F or a blue copy of H. Results of Łuczak, Ruciński and Voigt, and Kreuter determine the threshold for the property that the random graph G(n, p) is (F, H )v‐Ramsey. In this paper we consider the sister problem in the setting of randomly perturbed graphs. In particular, we determine

A probability measure on the subsets of the edge set of a graph G is a 1‐independent probability measure (1‐ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1‐ipm , denote by the associated random graph model. Let denote the collection of 1‐ipms on G for which each edge is included in with probability at least p. For , Balister and

Let T be a tree on n vertices and with maximum degree . We show that for the Glauber dynamics for k‐edge‐ colorings of T mixes in polynomial time in n. The bound on the number of colors is best possible as the chain is not even ergodic for . Our proof uses a recursive decomposition of the tree into subtrees; we bound the relaxation time of the original tree in terms of the relaxation time of its subtrees

We consider sufficient conditions for the existence of kth powers of Hamiltonian cycles in n‐vertex graphs G with minimum degree for arbitrarily small . About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that suffices for large n. For smaller values of the given graph G must satisfy

We study majority dynamics on the binomial random graph G(n, p) with p = d/n and , for some large . In this process, each vertex has a state in {− 1, + 1} and at each round every vertex adopts the state of the majority of its neighbors, retaining its state in the case of a tie. We show that with high probability the process reaches unanimity in at most four rounds. This confirms a conjecture of Benjamini

An edge‐ordered graph is a graph with a linear ordering of its edges. Two edge‐ordered graphs are equivalent if there is an isomorphism between them preserving the edge‐ordering. The edge‐ordered Ramsey number redge(H; q) of an edge‐ordered graph H is the smallest N such that there exists an edge‐ordered graph G on N vertices such that, for every q‐coloring of the edges of G, there is a monochromatic

Consider a complete graph Kn with edge weights drawn independently from a uniform distribution U(0, 1). The weight of the shortest (minimum‐weight) path P1 between two given vertices is known to be , asymptotically. Define a second‐shortest path P2 to be the shortest path edge‐disjoint from P1, and consider more generally the shortest path Pk edge‐disjoint from all earlier paths. We show that the cost

In this paper, we study corners in tree‐like and permutation tableaux. Tree‐like tableaux are in bijection with other combinatorial structures, including permutation tableaux, and have a connection to the partially asymmetric simple exclusion process (PASEP), an important model of an interacting particles system. In particular, in the context of tree‐like tableaux, a corner corresponds to a node occupied

Say that a graph G has property if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set and let e1, e2, … eN be a uniformly random ordering of the edges of Kn, with n an even integer. Let G0 be the empty graph on n vertices. For m ≥ 0, Gm + 1 is obtained from Gm by adding the edge em + 1 exactly if Gm ∪ {em + 1} has property . We analyze

Efficient algorithms for approximate counting and sampling in spin systems typically apply in the so‐called high‐temperature regime, where the interaction between neighboring spins is “weak.” Instead, recent work of Jenssen, Keevash, and Perkins yields polynomial‐time algorithms in the low‐temperature regime on bounded‐degree (bipartite) expander graphs using polymer models and the cluster expansion

Figure 1 Open in figure viewerPowerPoint

We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of vertices n. Moreover, we show the number of

The switch Markov chain has been extensively studied as the most natural Markov chain Monte Carlo approach for sampling graphs with prescribed degree sequences. We show that the switch chain for sampling simple undirected graphs with a given degree sequence is rapidly mixing when the degree sequence is so‐called strongly stable. Strong stability is satisfied by all degree sequences for which the switch

We study the local decodability and (tolerant) local testability of low‐degree n‐variate polynomials over arbitrary fields, evaluated over the domain {0,1}n. We show that for every field there is a tolerant local test whose query complexity depends only on the degree. In contrast we show that decodability is possible over fields of positive characteristic, but not over the reals.

Motivated by the celebrated Beck‐Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an discrepancy bound when n ≤ m and an O(1) bound when n ≫ mt. In this paper, we give a tight bound for the entire range of n and m, under a mild assumption that . The result is based on two steps

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a nonnegative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely

Let denote the power set of [n], ordered by inclusion, and let denote the random poset obtained from by retaining each element from independently at random with probability p and discarding it otherwise. Given any fixed poset F we determine the threshold for the property that contains F as an induced subposet. We also asymptotically determine the number of copies of a fixed poset F in . Finally, we

A 1‐factorization of a graph G is a collection of edge‐disjoint perfect matchings whose union is E (G ). In this paper, we prove that for any ϵ >0, an (n ,d ,λ )‐graph G admits a 1‐factorization provided that n is even, C 0 ≤ d  ≤ n −1 (where C 0=C 0(ϵ ) is a constant depending only on ϵ ), and λ  ≤ d 1−ϵ . In particular, since (as is well known) a typical random d ‐regular graph G n ,d is such a graph

Koiran's real τ ‐conjecture claims that the number of real zeros of a structured polynomial given as a sum of m products of k real sparse polynomials, each with at most t monomials, is bounded by a polynomial in mkt . This conjecture has a major consequence in complexity theory since it would lead to superpolynomial lower bounds for the arithmetic circuit size of the permanent. We confirm the conjecture

Two simple Markov processes are examined, one in discrete and one in continuous time, arising from idealized versions of a transmission protocol for mobile networks. We consider two independent walkers moving with constant speed on the discrete or continuous circle, and changing directions at independent geometric (respectively, exponential) times. One of the walkers carries a message that wishes to

We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on n vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when

We are concerned with the variance of a completely additive function defined on the symmetric group endowed with the Ewens probability. Overcoming specific dependence of the summands, we obtain the upper and lower bounds including optimal constants. We also derive a decomposition of such a function into a sum with uncorrelated summands. The results can be reformulated for the linear statistics defined

We study once‐reinforced random walk on . For this model, we derive limit results on all moments of its range using Tauberian theory.

We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons

We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic function s proved by Benjamini and Schramm. As a corollary we obtain that every planar nonamenable graph admits nonconstant Dirichlet harmonic function s27.

We consider the random‐cluster model (RCM) on with parameters p∈(0,1) and q ≥ 1. This is a generalization of the standard bond percolation (with edges open independently with probability p) which is biased by a factor q raised to the number of connected components. We study the well‐known Fortuin‐Kasteleyn (FK)‐dynamics on this model where the update at an edge depends on the global geometry of the

In this work we prove general bounds for the diameter of random graphs generated by a preferential attachment model whose parameter is a function f:N→[0,1] that drives the asymptotic proportion between the numbers of vertices and edges. These results are sharp when f is a regularly varying function at infinity with strictly negative index of regular variation −γ. For this particular class, we prove

Consider the Aldous Markov chain on the space of rooted binary trees with n labeled leaves in which at each transition a uniform random leaf is deleted and reattached to a uniform random edge. Now, fix 1 ≤ k