Rainbow Tur\'an problems for paths and forests of stars

TL;DR

This paper investigates the maximum edges in properly edge-colored graphs avoiding rainbow copies of specific forests of stars and paths, providing exact results for forests of stars and bounds for paths, challenging previous conjectures.

Contribution

It determines the rainbow Turán number exactly for forests of stars and offers new bounds for paths, advancing understanding of rainbow Turán problems.

Findings

Exact rainbow Turán number for forests of stars.

Bounds on rainbow Turán number for paths with k edges.

Disproof of a previous conjecture regarding paths.

Abstract

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a {\emph rainbow copy} of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the {\emph rainbow Tur\'an number} of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstra\"ete in 2007. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $k$ edges, disproving a conjecture in Keevash et al.