On the Rainbow Tur\'an number of paths

TL;DR

This paper investigates the maximum number of edges in edge-colored graphs that avoid rainbow paths of a certain length, providing improved upper bounds on the rainbow Turán number for paths.

Contribution

It establishes a tighter upper bound on the rainbow Turán number for paths, advancing the understanding of rainbow subgraph avoidance in edge-colored graphs.

Findings

Improved upper bound for rainbow Turán number of paths

Enhanced understanding of rainbow path avoidance in graphs

Provides a framework for future bounds on rainbow subgraphs

Abstract

Let $F$ be a fixed graph. The rainbow Tur\'an number of $F$ is defined as the maximum number of edges in a graph on $n$ vertices that has a proper edge-coloring with no rainbow copy of $F$ (where a rainbow copy of $F$ means a copy of $F$ all of whose edges have different colours). The systematic study of such problems was initiated by Keevash, Mubayi, Sudakov and Verstra\"ete. In this paper, we show that the rainbow Tur\'an number of a path with $k+1$ edges is less than $\left(\frac{9k}{7}+2\right) n$, improving an earlier estimate of Johnston, Palmer and Sarkar.