Comparable pairs in families of sets

TL;DR

This paper investigates the maximum number of comparable pairs in families of subsets of [n], resolving an old conjecture and providing bounds and characterizations for various family sizes.

Contribution

It resolves an old conjecture about the growth of comparable pairs and provides new bounds and characterizations for different family sizes.

Findings

Proves that c(n,m) = o(m^2) for certain m

Provides bounds for sparse and dense families

Characterizes extremal constructions for specific m values

Abstract

Given a family $\mathcal{F}$ of subsets of $[n]$, we say two sets $A, B \in \mathcal{F}$ are comparable if $A \subset B$ or $B \subset A$. Sperner's celebrated theorem gives the size of the largest family without any comparable 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 Erd\H{o}s 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 $c(n,m) = o(m^2)$ when $m = n^{\omega(1)} 2^{n/2}$. 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.