Extending the Erdős–Ko–Rado theorem

Let ${\cal F}$ be a k ‐uniform hypergraph on n vertices. Suppose that $|F_{1}\cap \cdots \cap F_{r}|\ge t$ holds for all $F_{1},\ldots ,F_{r}\in {\cal F}$ . We prove that the size of ${\cal F}$ is at most ${{n-t}\choose {k-t}}$ if $p= {k \over n}$ satisfies$${(1-p)}p^{{t\over {t+1}}(r-1)} - p^{{t}\over {t+1}}+p < 0$$ and n is sufficiently large. © 2005 Wiley Periodicals, Inc. J Combin Designs