Recently, the authors gave upper bounds for the size of 3-uniform hypergraphs avoiding a given odd cycle using the definition of a cycle due to Berge. In the present paper we extend this bound to m-uniform hypergraphs (for all m ≥ 3), as well as m-uniform hypergraphs avoiding a cycle of length 2k. Finally we consider non-uniform hypergraphs avoiding cycles of length 2k or 2k + 1. In both cases we can bound |h| by O(n1+1/k) under the assumption that all h ∈ ε() satisfy |h| ≥ 4k2.

In this paper, we consider pairs of forbidden subgraphs that imply the existence of a 2-factor in a graph. For d ≥ 2, let d be the set of connected graphs of minimum degree at least d. Let F1 and F2 be connected graphs and let be a set of connected graphs. Then {F1, F2} is said to be a forbidden pair for if every {F1, F2}-free graph in of sufficiently large order has a 2-factor. Faudree, Faudree and Ryjáček have characterized all the forbidden pairs for the set of 2-connected graphs. We first characterize the forbidden pairs for 2, which is a larger set than the set of 2-connected graphs, and observe a sharp difference between the characterized pairs and those obtained by Faudree, Faudree and Ryjáček. We then consider the forbidden pairs for connected graphs of large minimum degree. We prove that if {F1, F2} is a forbidden pair for d, then either F1 or F2 is a star of order at most d + 2. Ota and Tokuda have proved that every -free graph of minimum degree at least d has a 2-factor. These results imply that for k ≥ d + 2, no connected graphs F except for stars of order at most d + 2 make {K1,k, F} a forbidden pair for d, while for every connected graph F makes {K1,k, F} a forbidden pair for d. We consider the remaining range of , and prove that only a finite number of connected graphs F make {K1,k, F} a forbidden pair for d.

Let be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality || ≤ 2n−1. Suppose that ||=2n−1 + i. Choose the members of independently with probability p (delete them with probability 1 − p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all -element subsets. We determine the most probably inclusion-free family too, when the number of members is .

Given a locally finite connected infinite graph G, let the interval [pmin(G), pmax(G)] be the smallest interval such that if p > pmax(G), then every 1-independent bond percolation model on G with bond probability p percolates, and for p < pmin(G) none does. We determine this interval for trees in terms of the branching number of the tree. We also give some general bounds for other graphs G, in particular for lattices.

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.

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.

A family of sets (and the corresponding family of 0–1 vectors) is called t-cancellative if, for all distinct t + 2 members A1,. . ., At and B, C ∈ ,Let ct(n) be the size of the largest t-cancellative family on n elements, and let ct(n, r) denote the largest r-uniform family. We improve the previous upper bounds, e.g., we show c2(n) ≤ 20.322n (for n > n0). Using an algebraic construction we show that c2(n, 2k) = Θ(nk) for each k when n → ∞.

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 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.