Arrow relations on families of finite sets

Abstract

Let n, m and k be positive integers. Let X be a set of cardinality n, and let F be a family of subsets of X. We write ( n, m) → ( n − 1, m − k) when for all F with | F | ⩾ m, there exists an element x of X such that the family { F − { x} : F ϵ F } has cardinality at least m − k. We show that ( n, m)→( n − 1, m − 4) for all m ⩽ ⌈ 17 n/6⌉, n, m → ( n − 1, m − 5) for all m ⩽ ⌈13 n/4⌉, and ( n, m)→( n−1, m−6) for all m⩽⌈7 n/2⌉.