Rainbow Tur\'an number of clique subdivisions

TL;DR

This paper proves that sufficiently dense properly edge-coloured graphs necessarily contain rainbow subdivisions of complete graphs, extending classical clique subdivision results to the rainbow setting with sharp bounds.

Contribution

It introduces a method to find rainbow clique subdivisions in dense edge-coloured graphs, adapting the Sudakov-Tomon framework to the coloured context.

Findings

Dense properly edge-coloured graphs contain rainbow clique subdivisions.

The bound on the number of edges is sharp up to a small error.

The method extends existing frameworks to the coloured setting.

Abstract

We show that for any integer $t\geq 2$, every properly edge-coloured graph on $n$ vertices with more than $n^{1+o(1)}$ edges contains a rainbow subdivision of $K_t$. Note that this bound on the number of edges is sharp up to the $o(1)$ error term. This is a rainbow analogue of some classical results on clique subdivisions and extends some results on rainbow Tur\'an numbers. Our method relies on the framework introduced by Sudakov and Tomon[2020] which we adapt to find robust expanders in the coloured setting.