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arXiv · 2306.14682

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

Gennian Ge·Zixiang Xu·Yixuan Zhang
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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 $K_{5}$.

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Gennian Ge, Zixiang Xu, Yixuan Zhang. 2023-06-26. A new variant of the Erd\H{o}s-Gy\'{a}rf\'{a}s problem on $K_{5}$. https://arxiv.org/abs/2306.14682

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