Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels

In this paper we study conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erd\H{o}s on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum $d$-degree ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically determine the minimum vertex degree which guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also discuss an application to a problem of finding an optimal data allocation in a distributed storage system

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 Karpi\'nski, Ruci\'nski and Szyma\'nska; Szyma\'nska 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.Comment: 64 pages. Update includes minor revisions. To appear in Advances in Mathematic

Nonnegative k-sums, fractional covers, and probability of small deviations

More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that for any integers $n, k$ satisfying $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is also nonnegative. In this paper we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for $n \geq 33k^2$. This substantially improves the best previously known exponential lower bound $n \geq e^{ck \log\log k}$. In addition we prove a tight stability result showing that for every $k$ and all sufficiently large $n$, every set of $n$ reals with a nonnegative sum that does not contain a member whose sum with any other $k-1$ members is nonnegative, contains at least $\binom{n-1}{k-1}+\binom{n-k-1}{k-1}-1$ subsets of cardinality $k$ with nonnegative sum.Comment: 15 pages, a section of Hilton-Milner type result adde