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.

We study the lower tail large deviation problem for subgraph counts in a random graph. Let X H denote the number of copies of H in an Erdős–Rényi random graph $\mathcal{G}(n,p)$ . We are interested in estimating the lower tail probability $\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$ for fixed 0 < δ < 1.

Thanks to the results of Chatterjee, Dembo and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p ≥ n −α H (and conjecturally for a larger range of p). We study this variational problem and provide a partial characterization of the so-called ‘replica symmetric’ phase. Informally, our main result says that for every H, and 0 < δ < δH for some δH > 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite H and δ close to 1.

Judicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. Let k ≥ 2 be an integer and let G be a hypergraph with m i edges of size i for i=1,2. Bollobás and Scott conjectured that G has a partition into k classes, each of which contains at most $m_1/k+m_2/k^2+O(\sqrt{m_1+m_2})$ edges. In this paper, we confirm the conjecture affirmatively by showing that G has a partition into k classes, each of which contains at most

$$m_1/k+m_2/k^2+\ffrac{k-1}{2k^2}\sqrt{2(km_1+m_2)}+O(1)$$.

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, K d+1. As a corollary we find that K d+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 K d+1. This proves a conjecture of Galvin.

Let S be a set of n points in ${\mathbb R}^{2}$ contained in an algebraic curve C of degree d. We prove that the number of distinct distances determined by S is at least cdn 4/3, unless C contains a line or a circle.

We also prove the lower bound cd ′ min{m 2/3 n 2/3, m 2, n 2} for the number of distinct distances between m points on one irreducible plane algebraic curve and n points on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer and Solymosi in [19].

Let Dk denote the tournament on 3k vertices consisting of three disjoint vertex classes V 1, V 2 and V 3 of size k, each oriented as a transitive subtournament, and with edges directed from V 1 to V 2, from V 2 to V 3 and from V 3 to V 1. Fox and Sudakov proved that given a natural number k and ε > 0, there is n 0(k, ε) such that every tournament of order n ⩾ n 0(k,ε) which is ε-far from being transitive contains Dk as a subtournament. Their proof showed that $n_0(k,\epsilon ) \leq \epsilon ^{-O(k/\epsilon ^2)}$ and they conjectured that this could be reduced to n 0(k, ε) ⩽ ε−O(k). Here we prove this conjecture.

It is proved that the median eigenvalues of every connected bipartite graph G of maximum degree at most three belong to the interval [−1, 1] with a single exception of the Heawood graph, whose median eigenvalues are $\pm\sqrt{2}$ . Moreover, if G is not isomorphic to the Heawood graph, then a positive fraction of its median eigenvalues lie in the interval [−1, 1]. This surprising result has been motivated by the problem about HOMO-LUMO separation that arises in mathematical chemistry.

We study the distribution of products of conjugacy classes in finite simple groups, obtaining effective two-step mixing results, which give rise to an approximation to a conjecture of Thompson.

Our results, combined with work of Gowers and Viola, also lead to the solution of recent conjectures they posed on interleaved products and related complexity lower bounds, extending their work on the groups SL(2, q) to all (non-abelian) finite simple groups.

In particular it follows that, if G is a finite simple group, and A, B ⊆ Gt for t ⩾ 2 are subsets of fixed positive densities, then, as a = (a 1, . . ., at ) ∈ A and b = (b 1, . . ., bt ) ∈ B are chosen uniformly, the interleaved product a • b:=a 1 b 1 . . . atbt is almost uniform on G (with quantitative estimates) with respect to the ℓ∞-norm.

It also follows that the communication complexity of an old decision problem related to interleaved products of a, b ∈ Gt is at least Ω(t log |G|) when G is a finite simple group of Lie type of bounded rank, and at least Ω(t log log |G|) when G is any finite simple group. Both these bounds are best possible.

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 show that certain topologically defined uniform spanning tree probabilities for graphs embedded in an annulus can be computed as linear combinations of Pfaffians of matrices involving the line-bundle Green's function, where the coefficients count cover-inclusive Dyck tilings of skew Young diagrams.

We consider connectivity properties and asymptotic slopes for certain random directed graphs on ℤ2 in which the set of points $\mathcal{C}_o$ that the origin connects to is always infinite. We obtain conditions under which the complement of $\mathcal{C}_o$ has no infinite connected component. Applying these results to one of the most interesting such models leads to an improved lower bound for the critical occupation probability for oriented site percolation on the triangular lattice in two dimensions.

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 n λ(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.

We consider the problem of minimizing the number of triangles in a graph of given order and size, and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1−o(1))n 2.

We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.)

An important ingredient of our proofs is the following statement. Let S be a family of n open curves in ℝ2, so that each curve is the graph of a continuous real function defined on ℝ, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is $\Omega(nt\sqrt{\log t/\log\log t})$ .

Random increasing k-trees represent an interesting and useful class of strongly dependent graphs that have been studied widely, including being used recently as models for complex networks. In this paper we study an informative notion called BFS-profile and derive, by several analytic means, asymptotic estimates for its expected value, together with the limiting distribution in certain cases; some interesting consequences predicting more precisely the shapes of random k-trees are also given. Our methods of proof rely essentially on a bijection between k-trees and ordinary trees, the resolution of linear systems, and a specially framed notion called Flajolet–Odlyzko admissibility.