Let HG mean that every s-colouring of E(H) produces a monochromatic copy of G in some colour class. Let the s-colour degree Ramsey number of a graph G, written RΔ(G; s), be min{Δ(H): HG}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then RΔ(T; s) = s(k − 1) + ϵ, where ϵ = 1 when k is odd and ϵ = 0 when k is even. For general trees, RΔ(T; s) ≤ 2s(Δ(T) − 1).

To study sharpness of the upper bound, consider the double-starSa,b, the tree whose two non-leaf vertices have degrees a and b. If a ≤ b, then RΔ(Sa,b; 2) is 2b − 2 when a < b and b is even; it is 2b − 1 otherwise. If s is fixed and at least 3, then RΔ(Sb,b;s) = f(s)(b − 1) − o(b), where f(s) = 2s − 3.5 − O(s−1).

We prove several results about edge-colourings of bounded-degree graphs that are related to degree Ramsey numbers of paths. Finally, for cycles we show that RΔ(C2k + 1; s) ≥ 2s + 1, that RΔ(C2k; s) ≥ 2s, and that RΔ(C4;2) = 5. For the latter we prove the stronger statement that every graph with maximum degree at most 4 has a 2-edge-colouring such that the subgraph in each colour class has girth at least 5.

A rainbow subgraph of an edge-coloured graph is a subgraph whose edges have distinct colours. The colour degree of a vertex v is the number of different colours on edges incident with v. Wang and Li conjectured that for k ≥ 4, every edge-coloured graph with minimum colour degree k contains a rainbow matching of size at least ⌈k/2⌉. A properly edge-coloured K4 has no such matching, which motivates the restriction k ≥ 4, but Li and Xu proved the conjecture for all other properly coloured complete graphs. LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size ⌊k/2⌋ is guaranteed to exist, and they proved several sufficient conditions for a matching of size ⌈k/2⌉. We prove the conjecture in full.

Let G = G(d) be a random graph with a given degree sequence d, such as a random r-regular graph where r ≥ 3 is fixed and n = |G| → ∞. We study the percolation phase transition on such graphs G, i.e., the emergence as p increases of a unique giant component in the random subgraph G[p] obtained by keeping edges independently with probability p. More generally, we study the emergence of a giant component in G(d) itself as d varies. We show that a single method can be used to prove very precise results below, inside and above the ‘scaling window’ of the phase transition, matching many of the known results for the much simpler model G(n, p). This method is a natural extension of that used by Bollobás and the author to study G(n, p), itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.

A family of sets is said to be intersecting if A ∩ B ≠ ∅ for all A, B ∈ . It is a well-known and simple fact that an intersecting family of subsets of [n] = {1, 2, . . ., n} can contain at most 2n−1 sets. Katona, Katona and Katona ask the following question. Suppose instead ⊂ [n] satisfies || = 2n−1 + i for some fixed i > 0. Create a new family p by choosing each member of independently with some fixed probability p. How do we choose to maximize the probability that p is intersecting? They conjecture that there is a nested sequence of optimal families for i = 1, 2,. . ., 2n−1. In this paper, we show that the families [n](≥r) = {A ⊂ [n]: |A| ≥ r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values of i there exist optimal families lying between those we have found. It turns out that the optimal families we find simultaneously maximize the number of intersecting subfamilies of each possible order.

Standard compression techniques appear inadequate to solve the problem as they do not preserve intersection properties of subfamilies. Instead, our main tool is a novel compression method, together with a way of ‘compressing subfamilies’, which may be of independent interest.

We show that the list chromatic number of a simple d-regular r-uniform hypergraph is at least (1/2r log(2r2) + o(1)) log d if d is large.