We address two‐player combinatorial games whose graph of positions is a directed Galton–Watson tree. We consider normal and misère rules (where a player who cannot move loses or wins, respectively), as well as an “escape game” in which one designated player loses if either player cannot move. We study phase transitions for the probability of a draw or escape under optimal play, as the offspring distribution

Given a k‐vertex graph H and an integer n, what are the n‐vertex graphs with the maximum number of induced copies of H? This question is closely related to the inducibility problem introduced by Pippenger and Golumbic in 1975, which asks for the maximum possible fraction of k‐vertex subsets of an n‐vertex graph that induce a copy of H. Huang, Lee, and the first author proved that for a random k‐vertex

Frieze, Alan, Pegden, Wesley Random Structures & Algorithms, Vol. 53, No. 2, pp. 327– 351, September 2018 In “Online purchasing under uncertainty” we proved theorems concerning the cost of purchasing various combinatorial structures (paths, cycles, etc.) in graphs with randomly weighted edges, which are examined and either purchased or discarded one‐at‐a‐time by the purchaser. We discussed three models

We consider the generalized PageRank walk on a digraph G, with refresh probability and resampling distribution . We analyze convergence to stationarity when G is a large sparse random digraph with given degree sequences, in the limit of vanishing . We identify three scenarios: when is much smaller than the inverse of the mixing time of G the relaxation to equilibrium is dominated by the simple random

Let be the Erdős–Rényi graph with connection probability as N → ∞ for a fixed t ∈ (0, ∞). We derive a large‐deviations principle for the empirical measure of the sizes of all the connected components of , registered according to microscopic sizes (i.e., of finite order), macroscopic ones (i.e., of order N), and mesoscopic ones (everything in between). The rate function explicitly describes the microscopic

Consider the upper tail probability that the homomorphism count of a fixed graph H within a large sparse random graph Gn exceeds its expected value by a fixed factor . Going beyond the Erdős–Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random dn‐regular graphs (provided H has a regular 2‐core), and for sparse uniform random graphs. We further deal with joint upper

We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such permutations. We propose a technique to sample uniform permutations in such families as conditioned random colored walks. Building on that, we derive the behavior

In this paper we provide an algorithm that generates a graph with given degree sequence uniformly at random. Provided that , where is the maximal degree and m is the number of edges, the algorithm runs in expected time O(m). Our algorithm significantly improves the previously most efficient uniform sampler, which runs in expected time for the same family of degree sequences. Our method uses a novel

We consider the community detection problem in sparse random hypergraphs. Angelini et al. in [6] conjectured the existence of a sharp threshold on model parameters for community detection in sparse hypergraphs generated by a hypergraph stochastic block model. We solve the positive part of the conjecture for the case of two blocks: above the threshold, there is a spectral algorithm which asymptotically

We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity , can be viewed as a strong generalization of Jerrum

We study the behavior of random labeled and unlabeled cographs with n vertices as n tends to infinity. We show that both models admit a novel random graphon W1/2 as distributional limit. Our main tool is an enhanced skeleton decomposition of the random Pólya tree with n leaves and no internal vertices having only one child. As a byproduct, we obtain limits describing the asymptotic shape of this model

The basic random k‐SAT problem is: given a set of n Boolean variables, and m clauses of size k picked uniformly at random from the set of all such clauses on our variables, is the conjunction of these clauses satisfiable? Here we consider a variation of this problem where there is a bias towards variables occurring positive—that is, variables occur negated w.p. and positive otherwise—and study how

Starting from any graph on {1, … , n}, consider the Markov chain where at each time‐step a uniformly chosen vertex is disconnected from all of its neighbors and reconnected to another uniformly chosen vertex. This Markov chain has a stationary distribution whose support is the set of nonempty forests on {1, … , n}. The random forest corresponding to this stationary distribution has interesting connections

We study a new optimal stopping problem: Let G be a fixed graph with n vertices which become active on‐line in time, one by another, in a random order. The active part of G is the subgraph induced by the active vertices. Find a stopping algorithm that maximizes the expected number of connected components of the active part of G. We prove that if G is a k‐tree, then there is no asymptotically better

Given p ∈ (0, 1), we let be the random subgraph of the d‐dimensional hypercube Qd where edges are present independently with probability p. It is well known that, as d → ∞, if then with high probability Qp is connected; and if then with high probability Qp consists of one giant component together with many smaller components which form the “fragment”. Here we fix , and investigate the fragment, and

A coupling of random walkers on the same finite graph, who take turns sequentially, is said to be an avoidance coupling if the walkers never collide. Previous studies of these processes have focused almost exclusively on complete graphs, in particular how many walkers an avoidance coupling can include. For other graphs, apart from special cases, it has been unsettled whether even two noncolliding simple

At each vertex of , place a car with probability p or vacant parking spot with probability 1 − p. Cars perform independent random walks and park at vacant spots, rendering them passable. There is a transition at p = 1/2: the origin is a.s. visited by finitely many distinct cars when p < 1/2, and by infinitely many when p ≥ 1/2. Furthermore, a.s. all cars park if p ≤ 1/2 and some never park for p > 1/2

In the present paper we show that for any given digraph , that is, an oriented graph without self‐loops and 2‐cycles, one can construct a 1‐dependent Markov chain and n identically distributed hitting times T1, … , Tn on this chain such that the probability of the event Ti > Tj, for any i, j = 1, … , n, is larger than if and only if . This result is related to various paradoxes in probability theory

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

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