Abstract

Let $e(n,s)$ denote the maximum size of a family of subsets of an $n$-set without $s$ pairwise disjoint members. The problem of determining $e(n,s)$ was raised by Erdős more than half a century ago. Nevertheless except for the residue classes $n\equiv 0,-1,-2(\mathrm{mod}<!--FUNCTION\; APPLICATION-->s)$ the exact value of $e(n,s)$ is largely unknown. In the present paper $e(2s+r,s)$ is determined for most values of $1\le r\le s$ (see Theorems 1.8 and 1.9). The extremal families are so-called threshold families (see Definition 1.4).

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