m-Qualitatively independent families

Abstract

We say that two sets (A and B) are qualitatively independent if none of A ∩ B, A ∩ $\bar B$, Ā ∩ B, Ā ∩ $\bar B$ is empty, which means that they divide the underlying set into four parts. We show a short proof of the old result determining the maximum size of a pairwise qualitatively independent family on an n-element set, using the cyclic method of Katona. The cyclic method will be used for the general case, too, when we would like to determine the maximum size of a family on an n-element set for which it is required that there are two qualitatively independent among every m sets. However, it gives the exact result only when m is odd. For the case of even m our estimate is somewhat weaker than the expected exact value that we state in a form of a conjecture. Since our result is a special case of a difficult theorem of Gerbner (2013), this article can be considered as a new short proof for a special case.