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.