Intersection Properties of Subsets of Integers

Intersection properties of sets have been widely investigated by many authors. One type of theorems proved for them has the following form [9]. Let S be an n-element set and AI. ... , AN £ S, 1 £ [1, n]. Assume that IAi I1Ajl E 1 for 1,,;;;; i <j ,,;;;;N. How large can N be under this condition, depending on n and I? Thus, e.g., the de Bruijn-Erdos theorem [1] asserts that if IAi I1Ajl = 1 for all i ~ j, then N,,;;;; n. There are different extremal systems in the de Bruijn-Erdos theorem, but all they are known. Another typical example of intersection theorems is the well known result of Erdos, Ko and Rado [2]: If IAil = k and