An intersection graph of curves in the plane is called a string graph. Matoušek almost completely settled a conjecture of the authors by showing that every string graph with m edges admits a vertex separator of size $O(\sqrt{m}\log m)$. In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if Kt ⊈ G for some t, then the chromatic number of G is at most (log n)O(log t); (ii) if Kt,t ⊈ G, then G has at most t(log t)O(1)n edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdős–Hajnal conjecture almost holds for string graphs.

We propose a counting dimension for subsets of $\mathbb{Z}$ and prove that, under certain conditions on E,F ⊂ $\mathbb{Z}$, for Lebesgue almost every λ ∈ $\mathbb{R}$ the counting dimension of E + ⌊λF⌋ is at least the minimum between 1 and the sum of the counting dimensions of E and F. Furthermore, if the sum of the counting dimensions of E and F is larger than 1, then E + ⌊λF⌋ has positive upper Banach density for Lebesgue almost every λ ∈ $\mathbb{R}$. The result has direct consequences when E,F are arithmetic sets, e.g., the integer values of a polynomial with integer coefficients.

For any c ≥ 2, a c-strong colouring of the hypergraph G is an assignment of colours to the vertices of G such that, for every edge e of G, the vertices of e are coloured by at least min{c,|e|} distinct colours. The hypergraph G is t-intersecting if every two edges of G have at least t vertices in common.

A natural variant of a question of Erdős and Lovász is: For fixed c ≥ 2 and t ≥ 1, what is the minimum number of colours that is sufficient to c-strong colour any t-intersecting hypergraphs? The purpose of this note is to describe some open problems related to this question.

Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an exactly m-coloured complete subgraph for every natural number m or there exists an infinite subset X ⊂ $\mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex v in X such that X\{v} is 1-coloured and each edge between v and X\{v} has a distinct colour (all different to the colour used on X\{v}). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding exactly m-coloured complete subgraphs in colourings with finitely many colours.

Let G be a string graph (an intersection graph of continuous arcs in the plane) with m edges. Fox and Pach proved that G has a separator consisting of $O(m^{3/4}\sqrt{\log m})$ vertices, and they conjectured that the bound of $O(\sqrt m)$ actually holds. We obtain separators with $O(\sqrt m \,\log m)$ vertices.

Erdős asked in 1962 about the value of f(n,k,l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k,l)=(3,3) was solved by Lorden. Here we solve the problem (for all large n) for (3,l) with 4 ≤ l ≤ 7 and (k,3) with 4 ≤ k ≤ 7. Independently, Das, Huang, Ma, Naves and Sudakov resolved the cases (k,l)=(3,4) and (4,3).

Let {X1 , . . , Xn} be a collection of binary-valued random variables and let f : {0, 1}n → $\mathbb{R}$ be a Lipschitz function. Under a negative dependence hypothesis known as the strong Rayleigh condition, we show that f − ${\mathbb E}$f satisfies a concentration inequality. The class of strong Rayleigh measures includes determinantal measures, weighted uniform matroids and exclusion measures; some familiar examples from these classes are generalized negative binomials and spanning tree measures. For instance, any Lipschitz-1 function of the edges of a uniform spanning tree on vertex set V (e.g., the number of leaves) satisfies the Gaussian concentration inequality

\begin{linenomath}$${{\mathbb P} (f - {\mathbb E} f \geq a) \leq \exp \biggl( - \frac{a^2}{8 \, |V|} \biggr) }.$$\end{linenomath}

We consider numbers and sizes of independent sets in graphs with minimum degree at least d. In particular, we investigate which of these graphs yield the maximum numbers of independent sets of different sizes, and which yield the largest random independent sets. We establish a strengthened form of a conjecture of Galvin concerning the first of these topics.

A colouring of a graph G is called distinguishing if its stabilizer in Aut G is trivial. It has been conjectured that, if every automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We study properties of random 2-colourings of locally finite graphs and show that the stabilizer of such a colouring is almost surely nowhere dense in Aut G and a null set with respect to the Haar measure on the automorphism group. We also investigate random 2-colourings in several classes of locally finite graphs where the existence of a distinguishing 2-colouring has already been established. It turns out that in all of these cases a random 2-colouring is almost surely distinguishing.

For k-graphs F0 and H, an F0-packing of H is a family $\mathscr{F}$ of pairwise edge-disjoint copies of F0 in H. Let νF0(H) denote the maximum size |$\mathscr{F}$| of an F0-packing of H. Already in the case of graphs, computing νF0(H) is NP-hard for most fixed F0 (Dor and Tarsi [6]).

In this paper, we consider the case when F0 is a fixed linear k-graph. We establish an algorithm which, for ζ > 0 and a given k-graph H, constructs in time polynomial in |V(H)| an F0-packing of H of size at least νF0(H) − ζ |V(H)|k. Our result extends one of Haxell and Rödl, who established the analogous algorithm for graphs.

A minimum feedback arc set of a directed graph G is a smallest set of arcs whose removal makes G acyclic. Its cardinality is denoted by β(G). We show that a simple Eulerian digraph with n vertices and m arcs has β(G) ≥ m2/2n2+m/2n, and this bound is optimal for infinitely many m, n. Using this result we prove that a simple Eulerian digraph contains a cycle of length at most 6n2/m, and has an Eulerian subgraph with minimum degree at least m2/24n3. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find long cycles in Eulerian digraphs.