Let [n]r be the complete r-partite hypergraph with vertex classes of size n. It is an easy exercise to show that every set of more than (k−1)nr−1 edges in [n]r contains a matching of size k. We conjecture the following rainbow version of this observation: if F1,F2,. . .,Fk ⊆ [n]r are of size larger than (k−1)nr−1 then there exists a rainbow matching, that is, a choice of disjoint edges fi ∈ Fi. We prove this conjecture for r=2 and r=3.

We show that, for each fixed k, an n-vertex graph not containing a cycle of length 2k has at most $80\sqrt{k\log k}\cdot n^{1+1/k}+O(n)$ edges.

We consider the complete graph 𝜅n on n vertices with exponential mean n edge lengths. Writing Cij for the weight of the smallest-weight path between vertices i, j ∈ [n], Janson [18] showed that maxi,j∈[n]Cij/logn converges in probability to 3. We extend these results by showing that maxi,j∈[n]Cij − 3 logn converges in distribution to some limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure. Interestingly, this limiting random variable has also appeared as the weak limit of the re-centred graph diameter of the barely supercritical Erdős–Rényi random graph in [22].

We show that the maximum number of convex polygons in a triangulation of n points in the plane is O(1.5029n). This improves an earlier bound of O(1.6181n) established by van Kreveld, Löffler and Pach (2012), and almost matches the current best lower bound of Ω(1.5028n) due to the same authors. Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number of convex polygons in G.

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition with k parts if there exists an ε-regular partition with k parts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

We prove that triangular configurations are plentiful in large subsets of Cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving actions of a minimally almost periodic (not necessarily amenable or locally compact) group on a (not necessarily separable) probability space.

Consider the complete graph on n vertices, with edge weights drawn independently from the exponential distribution with unit mean. Janson showed that the typical distance between two vertices scales as log n/n, whereas the diameter (maximum distance between any two vertices) scales as 3 log n/n. Bollobás, Gamarnik, Riordan and Sudakov showed that, for any fixed k, the weight of the Steiner tree connecting k typical vertices scales as (k − 1)log n/n, which recovers Janson's result for k = 2. We extend this to show that the worst case k-Steiner tree, over all choices of k vertices, has weight scaling as (2k − 1)log n/n and finally, we generalize this result to Steiner trees with a mixture of typical and worst case vertices.

We consider the Widom–Rowlinson model of two types of interacting particles on d-regular graphs. We prove a tight upper bound on the occupancy fraction, the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on d + 1 vertices, Kd+1. As a corollary we find that Kd+1 also maximizes the normalized partition function of the Widom–Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalized number of homomorphisms from any d-regular graph G to the graph HWR, a path on three vertices with a loop on each vertex, is maximized by Kd+1. This proves a conjecture of Galvin.

Let $\mathcal F$ ⊂ 2[n] be a family of subsets. The diameter of $\mathcal F$ is the maximum of the size of symmetric differences among pairs of members of $\mathcal F$. In 1966 Kleitman determined the maximum of |$\mathcal F$| for fixed diameter. However, this important classical result lacked a characterization of the families meeting the bound. This is remedied in the present paper, where a best possible stability result is established as well.

In Section 4 we introduce a ‘parity trick’ that provides an easy way of deducing the odd case from the even case in both Kleitman's original theorem and its stability version.

Answering a question by Angel, Holroyd, Martin, Wilson and Winkler [1], we show that the maximal number of non-colliding coupled simple random walks on the complete graph KN, which take turns, moving one at a time, is monotone in N. We use this fact to couple [N/4] such walks on KN, improving the previous Ω(N/log N) lower bound of Angel et al. We also introduce a new generalization of simple avoidance coupling which we call partially ordered simple avoidance coupling, and provide a monotonicity result for this extension as well.

We identify the asymptotic probability of a configuration model CMn(d) producing a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which follows the asymptotic number of simple connected graphs with a prescribed degree sequence.

Let k ⩾ 3 be a fixed integer and let Zk(G) be the number of k-colourings of the graph G. For certain values of the average degree, the random variable Zk(G(n, m)) is known to be concentrated in the sense that $\tfrac{1}{n}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$ converges to 0 in probability (Achlioptas and Coja-Oghlan, Proc. FOCS 2008). In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees, $\tfrac{1}{\omega}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$ converges to 0 in probability for any diverging function $\omega=\omega(n)\ra\infty$. For k exceeding a certain constant k0 this result covers all average degrees up to the so-called condensation phase transitiondk,con, and this is best possible. As an application, we show that the experiment of choosing a k-colouring of the random graph G(n,m) uniformly at random is contiguous with respect to the so-called ‘planted model’.

Given a family of r-uniform hypergraphs ${\cal F}$ (or r-graphs for brevity), the Turán number ex(n,${\cal F})$ of ${\cal F}$ is the maximum number of edges in an r-graph on n vertices that does not contain any member of ${\cal F}$. A pair {u,v} is covered in a hypergraph G if some edge of G contains {u, v}. Given an r-graph F and a positive integer p ⩾ n(F), where n(F) denotes the number of vertices in F, let HFp denote the r-graph obtained as follows. Label the vertices of F as v1,. . .,vn(F). Add new vertices vn(F)+1,. . .,vp. For each pair of vertices vi, vj not covered in F, add a set Bi,j of r − 2 new vertices and the edge {vi, vj} ∪ Bi,j, where the Bi,j are pairwise disjoint over all such pairs {i, j}. We call HFp the expanded p-clique with an embedded F. For a relatively large family of F, we show that for all sufficiently large n, ex(n,HFp) = |Tr(n, p − 1)|, where Tr(n, p − 1) is the balanced complete (p − 1)-partite r-graph on n vertices. We also establish structural stability of near-extremal graphs. Our results generalize or strengthen several earlier results and provide a class of hypergraphs for which the Turán number is exactly determined (for large n).

We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of independent sets, among other combinatorial quantities. Our methods involve inequalities for determinants, for traces of functions of operators, and for entropy.

A subset C of edges in a k-uniform hypergraph H is a loose Hamilton cycle if C covers all the vertices of H and there exists a cyclic ordering of these vertices such that the edges in C are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random k-uniform hypergraph Hkn,p has vertex set [n] and an edge set E obtained by adding each k-tuple e ∈ ($\binom{[n]}{k}$) to E with probability p, independently at random.

Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but o(|E|) edges, referred to as the packing problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle in Hkn,p is

$p=\Theta\biggl(\frac{\log n}{n^{k-1}}\biggr),$

$N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$

Our proof utilizes and modifies the idea of ‘online sprinkling’ recently introduced by Vu and the first author.

In this paper we study a question related to the classical Erdős–Ko–Rado theorem, which states that any family of k-element subsets of the set [n] = {1,. . .,n} in which any two sets intersect has cardinality at most $\binom{n-1}{k-1}$.

We say that two non-empty families ${\mathcal A}, {\mathcal B}\subset \binom{[n]}{k}$ are s-cross-intersecting if, for any A ∈ ${\mathcal A}$, B ∈ ${\mathcal B}$, we have |A ∩ B| ≥ s. In this paper we determine the maximum of |${\mathcal A}$|+|${\mathcal B}$| for all n. This generalizes a result of Hilton and Milner, who determined the maximum of |${\mathcal A}$|+|${\mathcal B}$| for non-empty 1-cross-intersecting families.

We consider a random permutation drawn from the set of 132-avoiding permutations of length n and show that the number of occurrences of another pattern σ has a limit distribution, after scaling by nλ(σ)/2, where λ(σ) is the length of σ plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.

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.

We say a graph is (Qn,Qm)-saturated if it is a maximal Qm-free subgraph of the n-dimensional hypercube Qn. A graph is said to be (Qn,Qm)-semi-saturated if it is a subgraph of Qn and adding any edge forms a new copy of Qm. The minimum number of edges a (Qn,Qm)-saturated graph (respectively (Qn,Qm)-semi-saturated graph) can have is denoted by sat(Qn,Qm) (respectively s-sat(Qn,Qm)). We prove that

$$ \begin{linenomath} \lim_{n\to\infty}\ffrac{\sat(Q_n,Q_m)}{e(Q_n)}=0, \end{linenomath}$$

$$ \begin{linenomath} \ssat(Q_n,Q_m)\geq \ffrac{m+1}{2}\cdot 2^n, \end{linenomath}$$

We give an asymptotic formula for the minimum number of edges contained in triangles among graphs with n vertices and e edges. Our main tool is a generalization of Zykov's symmetrization method that can be applied to several graphs simultaneously.