New bounds on the size of nearly perfect matchings in almost regular hypergraphs

Let (Formula presented.) be a (Formula presented.) -uniform (Formula presented.) -regular simple hypergraph on (Formula presented.) vertices. Based on an analysis of the Rödl nibble, in 1997, Alon, Kim and Spencer proved that if (Formula presented.), then (Formula presented.) contains a matching covering all but at most (Formula presented.) vertices, and asked whether this bound is tight. In this paper we improve their bound by showing that for all (Formula presented.), (Formula presented.) contains a matching covering all but at most (Formula presented.) vertices for some (Formula presented.), when (Formula presented.) and (Formula presented.) are sufficiently large. Our approach consists of showing that the Rödl nibble process not only constructs a large matching but it also produces many well-distributed ‘augmenting stars’ which can then be used to significantly improve the matching constructed by the Rödl nibble process. Based on this, we also improve the results of Kostochka and Rödl from 1998 and Vu from 2000 on the size of matchings in almost regular hypergraphs with small codegree. As a consequence, we improve the best known bounds on the size of large matchings in combinatorial designs with general parameters. Finally, we improve the bounds of Molloy and Reed from 2000 on the chromatic index of hypergraphs with small codegree (which can be applied to improve the best known bounds on the chromatic index of Steiner triple systems and more general designs).