Rainbow Turán number of clique subdivisions

Abstract

We show that for any integer t ≥ 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 Turán numbers. Our method relies on the framework introduced by Sudakov and Tomon [10] which we adapt to find robust expanders in the coloured setting.

Robust (rainbow) subdivisions and simplicial cycles

“…Graphs with such expansion properties commonly appear in the study of sparse extremal problems, see e.g. [23,40,41,42]. Our definition immediately implies their existence in every graph with positive constant average degree, unlike earlier arguments.…”

“…A rainbow variant of the forbidden subdivison problem was recently proposed by Jiang, Methuku, and Yepremyan [23]. They proved that if a properly edge colored graph G on n vertices contains no rainbow subdivision of K t for some fixed t, then G has at most ne O( √ n) vertices.…”

Rainbow clique subdivisions and blow-ups

“…Das, Lee and Sudakov [2] answered the first question affirmatively, by showing that ex * (n, C) ≤ ne (log n) 1 2 +o (1) . Recently, Janzer [5] improved this bound by establishing that ex * (n, C) = O n(log n) 4 , which is tight up to a polylogarithmic factor. Very recently, Jiang, Methuku and Yepremyan [5] proved the following generalisation of Das, Lee and Sudakov [2] on ex * (n, C).…”

“…Theorem 1.1 (Jiang, Methuku, Yepremyan [5]). For every integer m ≥ 2 there exists a constant c > 0 such that for every integer n ≥ m the following holds.…”

“…The method used in [5] utilises robust expanders in the coloured setting together with a density increment argument, inspired in part by the method introduced by Sudakov and Tomon [12].…”