Rainbow Tur\'an problems for a matching and any other graph

TL;DR

This paper investigates the maximum sizes of edge collections in rainbow graphs that avoid certain subgraphs, focusing on minimum, sum, and product of edges in rainbow $$-free collections.

Contribution

It introduces bounds for the maximum edge counts in rainbow $$-free collections, extending Turán-type problems to rainbow graph settings.

Findings

Established bounds for minimum edge count in rainbow $$-free collections.

Derived maximum total edge sum for rainbow $$-free collections.

Analyzed the maximum product of edges in rainbow $$-free collections.

Abstract

For a family of graphs $\cF$, a graph is called $\cF$-free if it does not contain any member of $\cF$ as a subgraph. Given a collection of graphs $(G_1,\ldots,G_t)$ on the same vertex set $V$ of size $n$, a rainbow graph on $V$ is obtained by taking at most one edge from each $G_i$. We say that a collection is rainbow $\cF$-free if it contains no rainbow copy of any member of $\cF$. In this paper, we study the maximum values of $min_{i\in [t]}|E(G_i)|$, $\sum_{i=1}^{t}|E(G_i)|$ and $\prod_{i=1}^{t}|E(G_i)|$ among rainbow $\{F,M_{s+1}\}$-free collections $(G_1,\ldots,G_t)$ on $n$ vertices.