Largest component in Boolean sublattices

TL;DR

This paper extends Sperner's theorem by determining the maximum size of a family of subsets of an n-element set before the associated inclusion graph contains large components, using a reduction to rainbow cycle Turán problems.

Contribution

It provides an asymptotically sharp bound for the size of such families for intermediate component sizes, generalizing Sperner's theorem through a novel reduction approach.

Findings

Determines maximum size of family before large components appear in the inclusion graph.

Provides a sharp bound for when the inclusion graph becomes connected.

Links the problem to rainbow cycle Turán-type problems in edge-colored graphs.

Abstract

For a subfamily ${F}\subseteq 2^{[n]}$ of the Boolean lattice, consider the graph $G_{F}$ on ${F}$ based on the pairwise inclusion relations among its members. Given a positive integer $t$, how large can ${F}$ be before $G_{F}$ must contain some component of order greater than $t$? For $t=1$, this question was answered exactly almost a century ago by Sperner: the size of a middle layer of the Boolean lattice. For $t=2^n$, this question is trivial. We are interested in what happens between these two extremes. For $t=2^{g}$ with $g=g(n)$ being any integer function that satisfies $g(n)=o(n/\log n)$ as $n\to\infty$, we give an asymptotically sharp answer to the above question: not much larger than the size of a middle layer. This constitutes a nontrivial generalisation of Sperner's theorem. We do so by a reduction to a Tur\'an-type problem for rainbow cycles in properly edge-coloured graphs. Among other results, we also give a sharp answer to the question, how large can ${F}$ be before $G_{F}$ must be connected?