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

Let H $H$ be a k $k$ -uniform D $D$ -regular simple hypergraph on N $N$ vertices. Based an analysis of the Rödl nibble, in 1997, Alon, Kim and Spencer proved that if ⩾ 3 $k \geqslant 3$ , then contains matching covering all but at most − 1 / ( ) + o $ND^{-1/(k-1)+o(1)}$ vertices, asked whether this bound is tight. In paper we improve their by showing for > η $ND^{-1/(k-1)-\eta }$ vertices some = Θ 0 $\eta \Theta (k^{-3}) 0$ when are sufficiently large. Our approach consists nibble process not only constructs large it also produces many well-distributed ‘augmenting stars’ which can used to significantly constructed process. this, results Kostochka from 1998 Vu 2000 size matchings almost regular hypergraphs with small codegree. As consequence, best known bounds combinatorial designs general parameters. Finally, Molloy Reed chromatic index codegree (which applied Steiner triple systems more designs).