THE TURAN NUMBER OF BERGE-K4 IN TRIPLE SYSTEMS

A Berge-K-4 in a triple system is a configuration with four vertices v(1), v(2), v(3), v(4) and six distinct triples {e(ij) : 1 <= i < j <= 4} such that {v(i), v(j)} subset of e(ij) for every 1 <= i < j <= 4. We denote by B the set of Berge-K-4 configurations. A triple system is B-free if it does not contain any member of B. We prove that the maximum number of triples in a B-free triple system on n >= 6 points is obtained by the balanced complete 3-partite triple system: all triples {abc : a is an element of A, b is an element of B, c is an element of C} where A, B, C is a partition of n points with [n/3] = vertical bar A vertical bar <= vertical bar B vertical bar <= vertical bar C vertical bar = [n/3].