Cross-intersecting non-empty uniform subfamilies of hereditary families

Abstract

A set A t-intersects a set B if A and B have at least t common elements. A set of sets is called a family. Two families A and B are cross-the size of a smallest base of H. We show that for any integers t, r, and s with 1 ≤ t ≤ r ≤ s, there exists an integer c(r, s, t) such that the following holds for any hereditary family H with µ(H) ≥ c(r, s, t). If A is a non-empty subfamily of H (r) , B is a non-empty subfamily of H (s) , A and B are cross-t-intersecting, and |A| + |B| is maximum under the given conditions, then for some set I in H with t ≤ |I| ≤ r, either A = {A ∈ H (r) : I ⊆ A} and B = {B ∈ H (s) : |B ∩ I| ≥ t}, or r = s, t < |I|, A = {A ∈ H (r) : |A ∩ I| ≥ t}, and B = {B ∈ H (s) : I ⊆ B}. This was conjectured by the author for t = 1 and generalizes well-known results for the case where H is a power set.

Limits of eventual families of sets with application to algorithms for the common fixed point problem

“…We mention in passing that Borg [4] uses the term "hereditary family", in his work in the area of combinatorics, for exactly what we call here "co-eventual family". Several simple observations regarding such families can be made.…”