Rainbow Tur\'an number of clique subdivisions
Tao Jiang, Abhishek Methuku, Liana Yepremyan

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 , every properly edge-coloured graph on vertices with more than edges contains a rainbow subdivision of . Note that this bound on the number of edges is sharp up to the 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.
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