On the Size of K-Cross-Free Families

Two subsets A,B of an n-element ground set X are said to be crossing, if none of the four sets AB, A\B, B\A and X(AB) are empty. It was conjectured by Karzanov and Lomonosov forty years ago that if a family F of subsets of X does not contain k pairwise crossing elements, then |F|=O(kn). For k=2 and 3, the conjecture is true, but for larger values of k the best known upper bound, due to Lomonosov, is |F|=O(knlogn). In this paper, we improve this bound for large n by showing that |F|=O-k(nlogn) holds, where log denotes the iterated logarithm function.