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.
