On Large Systems of Sets with No Large Weak ∆-subsystems

The notion of a weak ∆-system was introduced and studied by Erdős, Milner and Rado [5] in 1974. A weak ∆-system is a family of sets where all pairs of sets have the same intersection size. Erdős and Szemerédi [7] investigated the behavior of the function F (n,r)—the largest integer so that there exists a family F of subsets of an n-element set which does not contain a ∆-system of r sets. Answering a question of Abbott, they proved that F (n,3) is superpolinomial in n: (1) F (n, 3) ≥ n log . They also conjectured that for some ε>0, F (n, 3) ≤ (2− ε). This conjecture was proved by Frankl and Rödl [8] for ε=0.01. Recently, Rödl and Thoma [9] substantially improved (1) by showing that for sufficiently large n,