Unavoidable Stars in 3-Graphs

Suppose F is a collection of 3-subsets of {1,2,…,n}. The problem of determining the least integer ƒ(n, k) with the property that if |F| > ƒ(n, k) then F 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, ƒ(n, k) = k(k − 1)n + O(k3) and, for k even, ƒ(n, k) = k(k − 32)n + O(n + k3).