arXiv subjects

Richard Mycroft

Publications and source records attributed to Richard Mycroft.

27 records · Page 2Linked to original sources

arXiv2015

A Multipartite Hajnal-Szemerédi Theorem

The celebrated Hajnal-Szemerédi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K_k-packing. Fischer's 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.

math.CO↗

arXiv2014

Tight cycles and regular slices in dense hypergraphs

We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of the original hypergraph. Accordingly we advocate their use in extremal hypergraph theory, and explain how they can lead to considerable simplifications in existing proofs in this field. We also use them for establishing the following two new results. Firstly, we prove a hypergraph extension of the Erdős-Gallai Theorem: for every $δ>0$ every sufficiently large $k$-uniform hypergraph with at least $(α+δ)\binom{n}{k}$ edges contains a tight cycle of length $αn$ for each $α\in[0,1]$. Secondly, we find (asymptotically) the minimum codegree requirement for a $k$-uniform $k$-partite hypergraph, each of whose parts has $n$ vertices, to contain a tight cycle of length $αkn$, for each $0<α<1$.

math.CO↗

arXiv2014

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 Karpiński, Ruciński and Szymańska; Szymańska 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.

math.CO↗

arXiv2014

A random version of Sperner's theorem

Let $\mathcal{P}(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal{P}(n,p)$ be obtained from $\mathcal{P}(n)$ by selecting elements from $\mathcal{P}(n)$ independently at random with probability $p$. A classical result of Sperner asserts that every antichain in $\mathcal{P}(n)$ has size at most that of the middle layer, $\binom{n}{\lfloor n/2 \rfloor}$. In this note we prove an analogous result for $\mathcal{P} (n,p)$: If $pn \rightarrow \infty$ then, with high probability, the size of the largest antichain in $\mathcal{P}(n,p)$ is at most $(1+o(1)) p \binom{n}{\lfloor n/2 \rfloor}$. This solves a conjecture of Osthus who proved the result in the case when $pn/\log n \rightarrow \infty$. 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 $p$.

math.CO↗

arXiv2013

A Geometric Theory for Hypergraph Matching

We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: 'space barriers' from convex geometry, and 'divisibility barriers' from arithmetic lattice-based constructions. To formulate precise results, we introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. We determine the essentially best possible minimum degree sequence for finding an almost perfect matching. Furthermore, our main result establishes the stability property: under the same degree assumption, if there is no perfect matching then there must be a space or divisibility barrier. This allows the use of the stability method in proving exact results. Besides recovering previous results, we apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemerédi Theorem. Here we prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical we defer it to a subsequent paper.

math.CO↗

arXiv2013

Hamilton l-cycles in uniform hypergraphs

We say that a k-uniform hypergraph C is an l-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 l vertices. We prove that if 1 \leq l \leq k-1 and k-l does not divide k then any k-uniform hypergraph on n vertices with minimum degree at least n/((\lceil (k/(k-l)) \rceil)(k-l))+o(n) contains a Hamilton l-cycle. This confirms a conjecture of Hàn and Schacht. Together with results of Rödl, Ruciński and Szemerédi, our result asymptotically determines the minimum degree which forces an l-cycle for any l with 1 \leq l \leq k-1.

math.CO↗

arXiv2010

Loose Hamilton cycles in hypergraphs

We prove that any k-uniform hypergraph on n vertices with minimum degree at least n/(2(k-1))+o(n) contains a loose Hamilton cycle. The proof strategy is similar to that used by Kühn 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.

math.CO↗

arXiv2010

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 $Δ$, any tournament on $(1+o(1))n$ vertices contains a copy of any directed tree on $n$ vertices with maximum degree at most $Δ$.

math.CO↗

arXiv2010

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

Sumner's universal tournament conjecture states that any tournament on $2n-2$ vertices contains any directed tree on $n$ vertices. In this paper we prove that this conjecture holds for all sufficiently large $n$. The proof makes extensive use of results and ideas from a recent paper by the same authors, in which an approximate version of the conjecture was proved.

math.CO↗