Perfect Matchings and K ³₄ -Tilings in Hypergraphs of Large Codegree

Abstract

For a k-graph F, let t _l (n, m, F) be the smallest integer t such that every k-graph G on n vertices in which every l-set of vertices is included in at least t edges contains a collection of vertex-disjoint F-subgraphs covering all but at most m vertices of G. Let K ^k _m denote the complete k-graph on m vertices.

The function $t_{k-1} (kn, 0, K_k^k)$ (i.e. when we want to guarantee a perfect matching) has been previously determined by Kühn and Osthus [9] (asymptotically) and by Rödl, Ruciński, and Szemerédi [13] (exactly). Here we obtain asymptotic formulae for some other l. Namely, we prove that for any $k \ge 4$ and $k/2 \le l \le k-2$,

$$ t_l(kn, 0, K_k^k) = \left(\frac{1}{2} + o(1)\right) {kn\choose k-l} $$

.

Also, we present various bounds in another special but interesting case: t ₂(n, m, K ³₄ ) with m = 0 or m = o(n), that is, when we want to tile (almost) all vertices by copies of K ³₄ , the complete 3-graph on 4 vertices.