Intersection theorems for systems of sets

On set systems not containing delta systems

“…As for exact values of r k), there is the trivial case 4(1, k) = k -1. And in [5], the case n = 2 was settled:…”

“…We mention two of these, from [5]. Let ~(n,k,m) denote the largest integer, s, for which there is a system of s n-sets not containing a A (k)-system, and not containing m pairwise disjoint sets.…”

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

“…Remark. Observe that for any r sets B(a (1) ),... ,B(a (r) ) in F , the vectors a (1) ,... ,a (r) can be reordered in such a way that h(a (1) , a (2) ) − h(a (1) , a (3) ) ≥ log 2 r − 1.…”

“…Let Q(a (1) , a (2) ) be the event that |C(a (1) , a (2) (1) , a (2) ). Let Q = (a (1) ,a (2) ) Q(a (1) , a (2) ).…”

“…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 n/4 log log n .…”