Delta-Systems and Qualitative (In)Dependence

Following Erdős and Rado, three sets are said to form a delta triple if any two of them have the same intersection. Let F(n,3) denote the largest cardinality of a family of subsets of an n-set not containing a delta-triple. It is not known whether lim supn→∞n−1 log F(n,3)<1. We say that a family of bipartitions of an n-set is qualitatively 3/4-weakly 3-dependent if the common refinement of any 3 distinct partitions of the family has at least 6 non-empty classes (i.e., at least 3/4 of the total). Let I(n) denote the maximum cardinality of such a family. We derive a simple relation between the exponential asymptotics of F(n,3) and I(n) and show, as a consequence, that lim supn→∞n−1 log F(n,3)=1 if and only if lim sup n→∞n−1 log I(n)=1.