We present a combinatorial lemma that provides a new approach to the two-sided exit problem and related questions for left-continuous random walks (i.e., random walks on the integers whose negative steps have size − 1). Some applications to random walks on the circle are also derived.

This paper describes methods for counting the number of nonnegative integer solutions of the system Ax = b when A is a nonnegative totally unimodular matrix and b an integral vector of fixed dimension. The complexity (under a unit cost arithmetic model) is strong in the sense that it depends only on the dimensions of A and not on the size of the entries of b. For the special case of ‘contingency tables’ the run-time is 2O(√dlogd) (where d is the dimension of the polytope). The method is complementary to Barvinok's in that our algorithm is effective on problems of high dimension with a fixed number of (non-sign) constraints, whereas Barvinok's algorithms are effective on problems of low dimension and an arbitrary number of constraints.

A set of integers is called a B2[g] set if every integer m has at most g representations of the form m = a + a′, with a ≤ a′ and a, a′ ∈ . We obtain a new lower bound for F(g, n), the largest cardinality of a B2[g] set in {1,. . .,n}. More precisely, we prove that infn→∞ where ϵg → 0 when g → ∞. We show a connection between this problem and another one discussed by Schinzel and Schmidt, which can be considered its continuous version.

A k-uniform tight cycle $C_s^k$ is a hypergraph on s > k vertices with a cyclic ordering such that every k consecutive vertices under this ordering form an edge. The pair (k, s) is admissible if gcd (k, s) = 1 or k / gcd (k,s) is even. We prove that if $s \ge 2{k^2}$ and H is a k-uniform hypergraph with minimum codegree at least (1/2 + o(1))|V(H)|, then every vertex is covered by a copy of $C_s^k$. The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order in which a tight path wraps around a complete k-partite k-uniform hypergraph, which may be of independent interest.

For hypergraphs F and H, a perfect F-tiling in H is a spanning collection of vertex-disjoint copies of F. For $k \ge 3$, there are currently only a handful of known F-tiling results when F is k-uniform but not k-partite. If s ≢ 0 mod k, then $C_s^k$ is not k-partite. Here we prove an F-tiling result for a family of non-k-partite k-uniform hypergraphs F. Namely, for $s \ge 5{k^2}$, every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V(H)| has a perfect $C_s^k$-tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph H such that $\mathrm{ex}(n,H)=\Theta(n^r)$ . So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$ , for integers $k\geq 2$ . In this paper, we add a new form of rationals for which the conjecture is true: $2-2/(2k+1)$ , for $k\geq 2$ . This in turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits $^{\prime}$ s conjecture, where one replaces a single graph by a finite family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits $^{\prime}$ s conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon $^{\prime}$ s conjecture. We also prove an upper bound on the Turán number of theta graphs in an asymmetric setting and employ this result to obtain another new rational exponent for Turán exponents: $r=7/5$ .

We consider an extension of the Monotone Subsequence lemma of Erdős and Szekeres in higher dimensions. Let v1,…,vn ∈ ℝd be a sequence of real vectors. For a subset I ⊆ [n] and vector [srarr ]c ∈ {0,1}d we say that I is [srarr ]c-free if there are no i < j ∈ I, such that, for every k = 1,…,d, vik < vik if and only if [srarr ]ck = 0. We construct sequences of vectors with the property that the largest [srarr ]c-free subset is small for every choice of [srarr ]c. In particular, for d = 2 the largest [srarr ]c-free subset is O(n⅝) for all the four possible [srarr ]c. The smallest possible value remains far from being determined.

We also consider and resolve a simpler variant of the problem.

Suppose that each vertex of a graph independently chooses a colour uniformly from the set {1, …, k}; and let Si be the random set of vertices coloured i. Farr shows that the probability that each set Si is stable (so that the colouring is proper) is at most the product of the k probabilities that the sets Si separately are stable. We give here a simple proof of an extension of this result.

We study a problem on edge percolation on product graphs G × K2. Here G is any finite graph and K2 consists of two vertices {0, 1} connected by an edge. Every edge in G × K2 is present with probability p independent of other edges. The bunkbed conjecture states that for all G and p, the probability that (u, 0) is in the same component as (v, 0) is greater than or equal to the probability that (u, 0) is in the same component as (v, 1) for every pair of vertices u, v ∈ G.

We generalize this conjecture and formulate and prove similar statements for randomly directed graphs. The methods lead to a proof of the original conjecture for special classes of graphs G, in particular outerplanar graphs.

Let Qd denote the hypercube of dimension d. Given d ⩾ m, a spanning subgraph G of Qd is said to be (Qd, Qm)-saturated if it does not contain Qm as a subgraph but adding any edge of E(Qd)\E(G) creates a copy of Qm in G. Answering a question of Johnson and Pinto [27], we show that for every fixed m ⩾ 2 the minimum number of edges in a (Qd, Qm)-saturated graph is Θ(2d).

We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of Qd is said to be weakly (Qd, Qm)-saturated if the edges of E(Qd)\E(G) can be added to G one at a time so that each added edge creates a new copy of Qm. Answering another question of Johnson and Pinto [27], we determine the minimum number of edges in a weakly (Qd, Qm)-saturated graph for all d ⩾ m ⩾ 1. More generally, we determine the minimum number of edges in a subgraph of the d-dimensional grid Pkd which is weakly saturated with respect to ‘axis aligned’ copies of a smaller grid Prm. We also study weak saturation of cycles in the grid.

We prove that the threshold for the appearance of a k-regular subgraph in Gn,p is at most the threshold for the appearance of a non-empty (k+1)-core. This improves a result of Pralat, Verstraete and Wormald [5] and proves a conjecture of Bollobás, Kim and Verstraete [3].

A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs. We give an alternative construction for such graphs.

A set S of permutations is forcing if for any sequence $\{\Pi_i\}_{i \in \mathbb{N}}$ of permutations where the density $d(\pi,\Pi_i)$ converges to $\frac{1}{|\pi|!}$ for every permutation $\pi \in S$ , it holds that $\{\Pi_i\}_{i \in \mathbb{N}}$ is quasirandom. Graham asked whether there exists an integer k such that the set of all permutations of order k is forcing; this has been shown to be true for any $k\ge 4$ . In particular, the set of all 24 permutations of order 4 is forcing. We provide the first non-trivial lower bound on the size of a forcing set of permutations: every forcing set of permutations (with arbitrary orders) contains at least four permutations.

We study the diameter of a family of random graphs on the torus that can be used to model wireless networks. In the random connection model two points x and y are connected with probability g(y−x), where g is a given function. We prove that the diameter of the graph is bounded by a constant, which depends only on ‖g‖1, with high probability as the number of vertices in the graph tends to infinity.

An [n, k, r]-partite graph is a graph whose vertex set, V, can be partitioned into n pairwise-disjoint independent sets, V1, …, Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 [les ] i < j [les ] n. An independent transversal in an [n, k, r]-partite graph is an independent set, T, consisting of n vertices, one from each Vi. An independent covering is a set of k pairwise-disjoint independent transversals. Let t(k, r) denote the maximal n for which every [n, k, r]-partite graph contains an independent transversal. Let c(k, r) be the maximal n for which every [n, k, r]-partite graph contains an independent covering. We give upper and lower bounds for these parameters. Furthermore, our bounds are constructive. These results improve and generalize previous results of Erdo″s, Gyárfás and Łuczak [5], for the case of graphs.

A graph construction game is a Maker–Breaker game. Maker and Breaker take turns in choosing previously unoccupied edges of the complete graph KN. Maker's aim is to claim a copy of a given target graph G while Breaker's aim is to prevent Maker from doing so. In this paper we show that if G is a d-degenerate graph on n vertices and N > d1122d+9n, then Maker can claim a copy of G in at most d1122d+7n rounds. We also discuss a lower bound on the number of rounds Maker needs to win, and the gap between these bounds.

We study random composite structures considered up to symmetry that are sampled according to weights on the inner and outer structures. This model may be viewed as an unlabelled version of Gibbs partitions and encompasses multisets of weighted combinatorial objects. We describe a general setting characterized by the formation of a giant component. The collection of small fragments is shown to converge in total variation toward a limit object following a Pólya–Boltzmann distribution.

Subgraph expansions are commonly used in the analysis of reliability measures of a failure-prone graph. We show that these expansions are special cases of a general result on the expected value of a random variable defined on a partially ordered set; when applied to random subgraphs, the general result defines a natural association between graph functions. As applications, we consider several graph invariants that measure the connectivity of a graph: the number of connected vertex sets of size k, the number of components of size k, and the total number of components. The expected values of these invariants on a random subgraph are global performance measures that generalize the ones commonly studied. Explicit results are obtained for trees, cycles, and complete graphs. Graphs which optimize these performance measures over a given class of graphs are studied

Let D1 and D2 be two dice with k and l integer faces, respectively, where k and l are two positive integers. The game Gn consists of tossing each die n times and summing the resulting faces. The die with the higher total wins the game. We examine the question of which die wins game Gn more often, for large values of n. We also give an example of a set of three dice which is non-transitive in game Gn for infinitely many values of n.

We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$ -regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ deviates from $\frac{n}{d+1}$ by at most $2$ . The second is that every graph on $n$ vertices with minimum degree $\delta$ contains a spanning subgraph in which the number of vertices of each degree does not exceed $\frac{n}{\delta +1}+2$ . Both conjectures remain open, but we prove several asymptotic relaxations for graphs with a large number of vertices $n$ . In particular we show that if $d^3 \log n \leq o(n)$ then every $d$ -regular graph with $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ is $(1+o(1))\frac{n}{d+1}$ . We also prove that any graph with $n$ vertices and minimum degree $\delta$ contains a spanning subgraph in which no degree is repeated more than $(1+o(1))\frac{n}{\delta +1}+2$ times.

The size-Ramsey number of a graph F is the smallest number of edges in a graph G with the Ramsey property for F, that is, with the property that any 2-colouring of the edges of G contains a monochromatic copy of F. We prove that the size-Ramsey number of the grid graph on n × n vertices is bounded from above by n3+o(1).