On the Maximum Number of Edges in Hypergraphs with Fixed Matching and Clique Number

Abstract

For a $k$-graph $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$, the clique number of $\mathcal{F}$ is defined to be the maximum size of a subset $Q$ of $[n]$ with $\left(\genfrac{}{}{0ex}{}{Q}{k}\right)\subset \mathcal{F}$. In the present paper, we determine the maximum number of edges in a $k$-graph on $[n]$ with matching number at most $s$ and clique number at least $q$ for $n\ge 8k^{2}s$ and for $q\ge (s+1)k-l$, $n\le (s+1)k+s/(3k)-l$. Two special cases that $q=(s+1)k-2$ and $k=2$ are solved completely.

26 pages