Abstract

For integers $n>k>1$ let $\left(\genfrac{}{}{0.0pt}{}{[n]}{k}\right)$ denote the collection of all $k$-subsets of the standard $n$-element set. For $s\ge 2$ and families ${\mathcal{ℱ}}_i\subset \left(\genfrac{}{}{0.0pt}{}{[n]}{k_i}\right)$, $1\le i\le s$, $(F_1,\dots <!--FUNCTION\; APPLICATION-->,F_s)$ is called a rainbow matching if $F_i\in {\mathcal{ℱ}}_i$ and the $F_i$ are pairwise disjoint. Theorem 1.5 provides the best possible upper bounds for the product of the sizes of ${\mathcal{ℱ}}_i$ if $n$ is sufficiently large and they span no rainbow matching. For the case of graphs $(k=2)$ some sharper bounds are established.

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