Rainbow copies of $F$ in families of $H$

TL;DR

This paper investigates the maximum number of copies of a graph $H$ in an $n$-vertex graph that avoids a rainbow $F$, providing exact solutions for specific cases and connecting to Turán and generalized Turán problems.

Contribution

It introduces a new extremal problem related to rainbow copies, solves the case when $H=F$ for certain graphs, and links the problem to Turán and generalized Turán theories.

Findings

Complete solution for $F$ as a 3-edge path when $H=F$

Asymptotic solution for $F$ as a book graph when $H=F$

Exact bounds for the maximum number of $H$ copies avoiding rainbow $F$

Abstract

We study the following problem. How many distinct copies of $H$ can an $n$-vertex graph $G$ have, if $G$ does not contain a rainbow $F$, that is, a copy of $F$ where each edge is contained in a different copy of $H$? The case $H=K_r$ is equivalent to the Tur\'an problem for Berge hypergraphs, which has attracted several researchers recently. We also explore the connection of our problem to the so-called generalized Tur\'an problems. We obtain several exact results. In the particularly interesting symmetric case where $H=F$, we completely solve the case $F$ is the 3-edge path, and asymptitically solve the case $F$ is a book graph.