A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilises Szemerédi's Regularity lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

We show that for each η > 0 every digraph G of sufficiently large order n is Hamiltonian if its out-and indegree sequences d +This gives an approximate solution to a problem of Nash- Williams (1975) [22] concerning a digraph analogue of Chvátal's theorem. In fact, we prove the stronger result that such digraphs G are pancyclic.

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In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large n:(i) [1-factorization conjecture] Suppose that n is even and D ≥ 2⌈n/4⌉ − 1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ ′ (G) = D. (ii) [Hamilton decomposition conjecture] Suppose that D ≥ ⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint Hamilton cycles in a graph of given minimum degree. According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer questions of Nash-Williams from 1970. The above bounds are best possible. In the current paper, we show the following: suppose that G is close to a complete balanced bipartite graph or to the union of two cliques of equal size. If we are given a suitable set of path systems which cover a set of 'exceptional' vertices and edges of G, then we can extend these path systems into an approximate decomposition of G into Hamilton cycles (or perfect matchings if appropriate).

Given positive integers k and ℓ where 4 divides k and k/2 ≤ ℓ ≤ k − 1, we give a minimum ℓ-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, as well as the hypergraph removal lemma and a structural result of Keevash and Sudakov relating to the Turán number of the expanded triangle.

Cameron and Erdős [6] asked whether the number of maximal sum-free sets in {1, . . . , n} is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2 ⌊n/4⌋ for the number of maximal sum-free sets. Here, we prove the following: For each 1 ≤ i ≤ 4, there is a constant C i such that, given any n ≡ i mod 4, {1, . . . , n} contains (C i + o(1))2 n/4 maximal sum-free sets. Our proof makes use of container and removal lemmas of Green [11,12], a structural result of Deshouillers, Freiman, Sós and Temkin [7] and a recent bound on the number of subsets of integers with small sumset by Green and Morris [13]. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.

Given positive integers k ≥ 3 and ℓ where k/2 ≤ ℓ ≤ k − 1, we give a minimum ℓ-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko [15] who gave an asymptotically exact result. Our approach makes use of the absorbing method, and builds on work in [21], where we proved the result for k divisible by 4.

Cameron and Erdős [6] raised the question of how many maximal sum-free sets there are in {1, . . . , n}, giving a lower bound of 2 ⌊n/4⌋ . In this paper we prove that there are in fact at most 2 (1/4+o(1))n maximal sum-free sets in {1, . . . , n}. Our proof makes use of container and removal lemmas of Green [8,9] as well as a result of Deshouillers, Freiman, Sós and Temkin [7] on the structure of sum-free sets.A fundamental notion in combinatorial number theory is that of a sum-free set: A set S of integers is sum-free if x + y ∈ S for every x, y ∈ S (note x and y are not necessarily distinct here). The topic of sum-free sets of integers has a long history. Indeed, in 1916 Schur [19] proved that, if n is sufficiently large, then any r-colouring of [n] := {1, . . . , n} yields a monochromatic triple x, y, z such that x + y = z.Note that both the set of odd numbers in [n] and the set {⌊n/2⌋ + 1, . . . , n} are maximal sum-free sets. (A sum-free subset of [n] is maximal if it is not properly contained in another sum-free subset of [n].) By considering all possible subsets of one of these maximal sum-free sets, we see that [n] contains at least 2 ⌈n/2⌉ sum-free sets. Cameron and Erdős [5] conjectured that in fact [n] contains only O(2 n/2 ) sum-free sets. The conjecture was proven independently by Green [8] and Sapozhenko [16]. Recently, a refinement of the Cameron-Erdős conjecture was proven in [1], giving an upper bound on the number of sum-free sets in [n] of size m (for each 1 ≤ m ≤ ⌈n/2⌉).Let f (n) denote the number of sum-free subsets of [n] and f max (n) denote the number of maximal sum-free subsets of [n]. Recall that the sum-free subsets of [n] described above lie in *