A new variant of the Erd\H{o}s-Gy\'{a}rf\'{a}s problem on $K_{5}$

TL;DR

This paper investigates a graph coloring problem related to extremal graph theory and coding theory, proving that a small number of colors suffice to avoid certain subgraphs with even color appearances, specifically for the complete graph on five vertices.

Contribution

The paper extends previous work on a variant of the Erdős-Gyárfás problem, providing a definitive answer for the case of $K_5$, which was previously unresolved.

Findings

Confirmed that a small number of colors suffices for $K_5$

Extended the problem from $K_4$ to $K_5$

Built on modified coloring functions from prior research

Abstract

Motivated by an extremal problem on graph-codes that links coding theory and graph theory, Alon recently proposed a question aiming to find the smallest number $t$ such that there is an edge coloring of $K_{n}$ by $t$ colors with no copy of given graph $H$ in which every color appears an even number of times. When $H=K_{4}$, the question of whether $n^{o(1)}$ colors are enough, was initially emphasized by Alon. Through modifications to the coloring functions originally designed by Mubayi, and Conlon, Fox, Lee and Sudakov, the question of $K_{4}$ has already been addressed. Expanding on this line of inquiry, we further study this new variant of the generalized Ramsey problem and provide a conclusively affirmative answer to Alon's question concerning $K_{5}$.