Colouring set families without monochromatic k-chains

TL;DR

This paper investigates coloured extremal problems in set families, extending classical theorems like Sperner's and Erdős' to include constraints on monochromatic chains, and identifies structures that maximize colourings.

Contribution

It extends Erdős-Rothschild type problems to set families, proving which structures maximize the number of colourings avoiding monochromatic chains for large n.

Findings

Largest $k$-chain-free families maximize $(2,k)$-colourings for large n.

Middle level families maximize $(3,2)$-colourings.

Asymptotic results for maximum $(r,k)$-colourings when $r(k-1)$ divisible by three.

Abstract

A coloured version of classic extremal problems dates back to Erd\H{o}s and Rothschild, who in 1974 asked which $n$-vertex graph has the maximum number of 2-edge-colourings without monochromatic triangles. They conjectured that the answer is simply given by the largest triangle-free graph. Since then, this new class of coloured extremal problems has been extensively studied by various researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of Sperner's Theorem, the classic result in extremal set theory on the size of the largest antichain in the Boolean lattice, and Erd\H{o}s' extension to $k$-chain-free families. Given a family $\mathcal{F}$ of subsets of $[n]$, we define an $(r,k)$-colouring of $\mathcal{F}$ to be an $r$-colouring of the sets without any monochromatic $k$-chains $F_1 \subset F_2 \subset \dots \subset F_k$. We prove that for $n$ sufficiently large in terms of $k$, the largest $k$-chain-free families also maximise the number of $(2,k)$-colourings. We also show that the middle level, $\binom{[n]}{\lfloor n/2 \rfloor}$, maximises the number of $(3,2)$-colourings, and give asymptotic results on the maximum possible number of $(r,k)$-colourings whenever $r(k-1)$ is divisible by three.