Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings

TL;DR

This paper generalizes anti-Ramsey theory by introducing functions that determine minimal or maximal colorings in edge-colored complete graphs to force or avoid monochromatic subgraphs with specific properties, providing asymptotic results.

Contribution

It introduces the $f$ and $g$ functions for generalized anti-Ramsey problems and develops methods for asymptotic analysis across various graph families.

Findings

Derived asymptotically tight bounds for the $f$-function.

Derived asymptotically tight bounds for the $g$-function.

Established methods applicable to many graph families.

Abstract

Anti-Ramsey theory was initiated in 1975 by Erd\H{o}s, Simonovits and S\'os, inspiring hundreds of publications since then. The present work is the third and last piece of our trilogy in which we introduce a far-reaching generalization via the following two functions for any graph $G$ and family ${\cal F}$ of graphs: If $K_2 \in {\cal F}$, let $f(n,G|{\cal F})$ be the smallest integer $k$ such that every edge coloring of $K_n$ with at least $k$ colors forces a copy of $G$ in which all color classes are members of ${\cal F}$. If $K_2 \notin {\cal F}$, let $g(n,G|{\cal F})$ be the largest integer $k$ for which there exists an edge coloring of $K_n$ using exactly $k$ colors, such that every copy of $G$ contains an induced color class which is a member of ${\cal F}$. We develop methods suitable for deriving asymptotically tight results for the $f$-function and the $g$-function for many combinations of $G$ and ${\cal F}$. The preceding parts of the trilogy are arXiv: 2405.19812 and 2408.04257, published in Discrete Applied Math. Vol. 363 and Mathematics Vol. 12:23, respectively.