Colourings without monochromatic disjoint pairs

TL;DR

This paper develops a unified approach to the Erd ext{o}s-Rothschild problem for intersecting structures, extending previous results and characterizing families of vector spaces that maximize colourings without monochromatic disjoint pairs.

Contribution

It introduces a unified method for Erd ext{o}s-Rothschild problems in intersecting structures, extending bounds and characterizing extremal families.

Findings

Extended bounds on the size of the ground set for intersecting structures.

Characterized extremal families of vector spaces asymptotically maximizing colourings.

Addressed a conjecture of Hoppen, Lefmann and Odermann.

Abstract

The typical extremal problem asks how large a structure can be without containing a forbidden substructure. The Erd\H{o}s-Rothschild problem, introduced in 1974 by Erd\H{o}s and Rothschild in the context of extremal graph theory, is a coloured extension, asking for the maximum number of colourings a structure can have that avoid monochromatic copies of the forbidden substructure. The celebrated Erd\H{o}s-Ko-Rado theorem is a fundamental result in extremal set theory, bounding the size of set families without a pair of disjoint sets, and has since been extended to several other discrete settings. The Erd\H{o}s-Rothschild extensions of these theorems have also been studied in recent years, most notably by Hoppen, Koyakayawa and Lefmann for set families, and Hoppen, Lefmann and Odermann for vector spaces. In this paper we present a unified approach to the Erd\H{o}s-Rothschild problem for intersecting structures, which allows us to extend the previous results, often with sharp bounds on the size of the ground set in terms of the other parameters. In many cases we also characterise which families of vector spaces asymptotically maximise the number of Erd\H{o}s-Rothschild colourings, thus addressing a conjecture of Hoppen, Lefmann and Odermann.