Let H be a k-uniform hypergraph on n vertices where n is a sufficiently large integer not divisible by k. We prove that if the minimum (k − 1)-degree of H is at least ⌊n/k⌋, then H contains a matching with ⌊n/k⌋ edges. This confirms a conjecture of Rödl, Ruciński and Szemerédi [13], who proved that minimum (k − 1)-degree n/k + O(log n) suffices. More generally, we show that H contains a matching of size d if its minimum codegree is d < n/k, which is also best possible.

In this paper we study the following problem.

Discrete partitioning problem (DPP). Let $\mathbb{F}_q$Pn denote the n-dimensional finite projective space over $\mathbb{F}_q$. For positive integer k ⩽ n, let {Ai}i = 1N be a partition of ($\mathbb{F}_q$Pn)k such that:

1. (1) for all i ⩽ N, Ai = ∏j=1kAji (partition into product sets),

2. (2) for all i ⩽ N, there is a (k − 1)-dimensional subspace Li ⊆ $\mathbb{F}_q$Pn such that Ai ⊆ (Li)k.

DPP arises in an approach that we propose for proving lower bounds for the query complexity of generating random points from convex bodies. It is also related to other partitioning problems in combinatorics and complexity theory. We conjecture an asymptotically optimal partition for DPP and show that it is optimal in two cases: when the dimension is low (k = n = 2) and when the factors of the parts are structured, namely factors of a part are close to being a subspace. These structured partitions arise naturally as partitions induced by query algorithms. Our problem does not seem to be directly amenable to previous techniques for partitioning lower bounds such as rank arguments, although rank arguments do lie at the core of our techniques.

Hajnal and Szemerédi proved that every graph G with |G| = ks and δ(G)⩾ k(s − 1) contains k disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph $\vv G$ with |$\vv G$| = ks and δ($\vv G$) ⩾ 2k(s − 1) − 1 contains k disjoint transitive tournaments on s vertices, where δ($\vv G$)= minv∈V($\vv G$)d−(v)+d+(v). Our result implies the Hajnal–Szemerédi theorem, and its degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.

Let h > w > 0 be two fixed integers. Let H be a random hypergraph whose hyperedges are all of cardinality h. To w-orient a hyperedge, we assign exactly w of its vertices positive signs with respect to the hyperedge, and the rest negative signs. A (w,k)-orientation of H consists of a w-orientation of all hyperedges of H, such that each vertex receives at most k positive signs from its incident hyperedges. When k is large enough, we determine the threshold of the existence of a (w,k)-orientation of a random hypergraph. The (w,k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when h = 2 and w = 1, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran. This settled a conjecture of Karp and Saks.

The shadow of a system of sets is all sets which can be obtained by taking a set in the original system, and removing a single element. The Kruskal-Katona theorem tells us the minimum possible size of the shadow of $\mathcal A$, if $\mathcal A$ consists of m r-element sets.

In this paper, we ask questions and make conjectures about the minimum possible size of a partial shadow for $\mathcal A$, which contains most sets in the shadow of $\mathcal A$. For example, if $\mathcal B$ is a family of sets containing all but one set in the shadow of each set of $\mathcal A$, how large must $\mathcal B$ be?