Richard Mycroft

Loose Hamilton cycles in hypergraphs

We prove that any k-uniform hypergraph on n vertices with minimum degree at least n2(k-1)+o(n) contains a loose Hamilton cycle. The proof strategy is similar to that used by Kuhn and Osthus for the 3-uniform case. Though some additional difficulties arise in the k-uniform case, our argument here is considerably simplified by applying the recent hypergraph blow-up lemma of Keevash.

Hamilton ℓ-cycles in uniform hypergraphs

We say that a k-uniform hypergraph C is an @?-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely @? vertices. We prove that if 1=<@?

Polynomial-time perfect matchings in dense hypergraphs

Let H be a k-graph on n vertices, with minimum codegree at least n/k + cn for some fixed c > 0. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in H or a certificate that none exists. This essentially solves a problem of Karpinski, Rucinski and Szymanska, who previously showed that this problem is NP-hard for a minimum codegree of n/k - cn. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.

Packing k-partite k-uniform hypergraphs

Let G and H be k-graphs (k-uniform hypergraphs); then a perfect H-packing in G is a collection of vertex-disjoint copies of H in G which together cover every vertex of G. For any fixed H let ? ( H , n ) be the minimum ? such that any k-graph G on n vertices with minimum codegree ? ( G ) ? ? contains a perfect H-packing. The problem of determining ? ( H , n ) has been widely studied for graphs (i.e. 2-graphs), but little is known for k ? 3 . Here we determine the asymptotic value of ? ( H , n ) for all complete k-partite k-graphs H, as well as a wide class of other k-partite k-graphs. In particular, these results provide an asymptotic solution to a question of Rodl and Rucinski on the value of ? ( H , n ) when H is a loose cycle. We also determine asymptotically the codegree threshold needed to guarantee an H-packing covering all but a constant number of vertices of G for any complete k-partite k-graph H.

An approximate version of Sumner's universal tournament conjecture

Sumner@?s universal tournament conjecture states that any tournament on 2n-2 vertices contains a copy of any directed tree on n vertices. We prove an asymptotic version of this conjecture, namely that any tournament on (2+o(1))n vertices contains a copy of any directed tree on n vertices. In addition, we prove an asymptotically best possible result for trees of bounded degree, namely that for any fixed @D, any tournament on (1+o(1))n vertices contains a copy of any directed tree on n vertices with maximum degree at most @D.

A multipartite Hajnal-Szemerédi theorem

The celebrated Hajnal-Szemeredi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K k -packing. Fischers conjecture states that the analogous result holds for all multipartite graphs except for those formed by a single construction. Recently, we deduced an approximate version of this conjecture from new results on perfect matchings in hypergraphs. In this paper, we apply a stability analysis to the extremal cases of this argument, thus showing that the exact conjecture holds for any sufficiently large graph.

A random version of Sperner's theorem

Let P ( n ) denote the power set of n , ordered by inclusion, and let P ( n , p ) be obtained from P ( n ) by selecting elements from P ( n ) independently at random with probability p. A classical result of Sperner 12 asserts that every antichain in P ( n ) has size at most that of the middle layer, ( n ? n / 2 ? ) . In this note we prove an analogous result for P ( n , p ) : If p n ? ∞ then, with high probability, the size of the largest antichain in P ( n , p ) is at most ( 1 + o ( 1 ) ) p ( n ? n / 2 ? ) . This solves a conjecture of Osthus 9 who proved the result in the case when p n / log ? n ? ∞ . Our condition on p is best-possible. In fact, we prove a more general result giving an upper bound on the size of the largest antichain for a wider range of values of p.

A proof of Sumner's universal tournament conjecture for large tournaments

Sumners universal tournament conjecture states that any tournament on

Tight cycles and regular slices in dense hypergraphs

2n-2

The minimum vertex degree for an almost-spanning tight cycle in a 3-uniform hypergraph

vertices contains any directed tree on