Unavoidable Stars in 3-Graphs

Suppose .F is a collection of 3-subsets of (1,2,..., n). The problem of determining the least integer f(n, k) with the property that if l,FT/ >f(n, k) then .7 contains a k-star (i.e., k 3-sets such that the intersection of any pair of them consists of exactly the same element) is studied. It is proved that, for k odd, f(n, k) = k(k 1) n + P(k-‘) and, for k even,f(n, k) = k(k 3/2) n + F(n + k’).