Comparable pairs in families of sets

Abstract

Given a family F of subsets of [n], we say two sets A,B ∈ F are comparable if A ⊂ B or B ⊂ A. Sperner’s celebrated theorem gives the size of the largest family without any compa-rable pairs. This result was later generalised by Kleitman, who gave the minimum number of comparable pairs appearing in families of a given size. In this paper we study a complementary problem posed by Erdős and Daykin and Frankl in the early ’80s. They asked for the maximum number of comparable pairs that can appear in a family of m subsets of [n], a quantity we denote by c(n,m). We first resolve an old conjecture of Alon and Frankl, showing that c(n,m) = o(m2) when m = nω(1)2n/2. We also obtain more accurate bounds for c(n,m) for sparse and dense families, characterise the extremal constructions for certain values of m, and sharpen some other known results.