On r-wise t-intersecting Uniform Families

Abstract

We consider families, $\mathcal {F}$ of k-subsets of an n-set. For integers $r\ge 2$, $t\ge 1$, $\mathcal {F}$ is called r-wise t-intersecting if any r of its members have at least t elements in common. The most natural construction of such a family is the full t-star, consisting of all k-sets containing a fixed t-set. In the case $r=2$ the Exact Erdős-Ko-Rado Theorem shows that the full t-star is largest if $n\ge (t+1)(k-t+1)$. In the present paper, we prove that for $n\ge (2.5t)^{1/(r-1)}(k-t)+k$, the full t-star is largest in case of $r\ge 3$. Examples show that the exponent $\frac{1}{r-1}$ is best possible. This represents a considerable improvement on a recent result of Balogh and Linz.