# 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.

## Key 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.