On hypergraphs without two edges intersecting in a given number of vertices

Abstract

Let X be a finite set of n-melements and suppose t ⩾ 0 is an integer. In 1975, P. Erdös asked for the determination of the maximum number of sets in a family F = { F 1,…, F m }, F i ⊂ X, such that ∥ F i ∩ F j ∥ ≠ t for 1 ⩽ i ≠ j ⩽ m. This problem is solved for n ⩾ n 0( t). Let us mention that the case t = 0 is trivial, the answer being 2 n − 1 . For t = 1 the problem was solved in [3]. For the proof a result of independent interest (Theorem 1.5) is used, which exhibits connections between linear algebra and extremal set theory.