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.

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.

We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability $\frac12$. This random graph model is denoted $G(n,\frac12,k)$. The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov [3] uses spectral techniques to find the hidden clique with high probability when $k = c \sqrt{n}$ for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron [12]. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n2). However, the analysis in [12] gives a success probability of only 2/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time O(n2) (that is, linear in the size of the input) and has a failure probability that tends to 0 as n tends to ∞. We develop this algorithm in the more general setting where the clique is replaced by a dense random graph.

We show that asymptotically almost surely a tree with m edges decomposes the complete bipartite graph K2m,2m, a result connected to a conjecture of Graham and Häggkvist. The result also implies that asymptotically almost surely a tree with m edges decomposes the complete graph with O(m2) edges. An ingredient of the proof consists in showing that the bipartition classes of the base tree of a random tree have roughly equal size.

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 study the (1:b) Maker–Breaker component game, played on the edge set of a d-regular graph. Maker's aim in this game is to build a large connected component, while Breaker's aim is to prevent him from doing so. For all values of Breaker's bias b, we determine whether Breaker wins (on any d-regular graph) or Maker wins (on almost every d-regular graph) and provide explicit winning strategies for both players.

To this end, we prove an extension of a theorem of Gallai, Hasse, Roy and Vitaver about graph orientations without long directed simple paths.

We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-ε)Δt for graphs of maximum degree at most Δ, where ε is some absolute positive constant independent of t. The other is a bound of O(Δt/log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance-t chromatic index at least Ω(Δt/log Δ).

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.

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.

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.

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}