Delta-systems and qualitative (in)dependence (Q5917474)

Denote \(F(n,3)\) the largest cardinality of a family of subsets of an \(n\)-set not containing a delta-triple. Due to the well-known Erdős-Rado theorem, this is a well-defined, finite quantity, but it is unknown whether \(\limsup_{n\rightarrow \infty} n^{-1}\log F(n,3) < 1.\) The paper introduces the notion of qualitative 3/4-weakly 3-dependence of a family of bipartitions of an \(n\)-set: the common refinements of any 3 distinct bipartitions must contain at least 6 non-empty classes (i.e. at least 3/4 of the total). Denoting by \(I(n)\) the maximum cardinality of such a family, the authors derive a simple relation between the exponential asymptotics of \(F(n,3)\) and \(I(n)\) and they show that \(\limsup_{n\rightarrow \infty} n^{-1}\log F(n,3) = 1\) if and only if \(\limsup_{n\rightarrow \infty} n^{-1}\log I(n) = 1.\)