Proof of the Erdős matching conjecture in a new range

Let s > k ≧ 2 be integers. It is shown that there is a positive real e = e(k) such that for all integers n satisfying (s + 1)k ≦ n < (s + 1)(k + e) every k-graph on n vertices with no more than s pairwise disjoint edges has at most $$\left( {\begin{array}{*{20}{c}} {\left( {s + 1} \right)k - 1} \\ k \end{array}} \right)$$ edges in total. This proves part of an old conjecture of Erdős.