Rainbow Turán number of even cycles

Abstract: The rainbow Tur\'an number $\mathrm{ex}^*(n,H)$ of a graph $H$ is the maximum possible number of edges in a properly edge-coloured $n$-vertex graph with no rainbow subgraph isomorphic to $H$. We prove that for any integer $k\geq 2$, $\mathrm{ex}^*(n,C_{2k})=O(n^{1+1/k})$. This is tight and establishes a conjecture of Keevash, Mubayi, Sudakov and Verstra\"ete. We use the same method to prove several other conjectures in various topics. For example, we give an upper bound for the Tur\'an number of the blow-ups of even cycles, which can be used to disprove a conjecture of Erd\H os and Simonovits.

combinatorics

Audience: researchers in the topic

Comments: password 111317

SCMS Combinatorics Seminar

Series comments: Check scmscomb.github.io/ for more information