Multicolor Tur\'an numbers

TL;DR

This paper investigates a generalized Turán problem involving edge-disjoint copies of a fixed graph and forbidden subgraphs, providing asymptotic results and characterizing certain pairs of graphs where the maximum number is quadratic.

Contribution

It characterizes pairs of graphs for which the maximum number of edge-disjoint copies is quadratic and develops asymptotic bounds using advanced combinatorial tools.

Findings

Identifies pairs {F, G} with quadratic order of magnitude for ex_F(n,G)

Provides asymptotic bounds for edge-disjoint graph packings

Utilizes regularity lemma, supersaturation, and graph packing techniques

Abstract

We consider a natural generalisation of Tur\'an's forbidden subgraph problem and the Ruzsa-Szemer\'edi problem by studying the maximum number $ex_F(n,G)$ of edge-disjoint copies of a fixed graph $F$ can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. We determine the pairs $\{F, G\}$ for which the order of magnitude of $ex_F(n,G)$ is quadratic and prove several asymptotic results using various tools from the regularity lemma and supersaturation to graph packing results.