Rainbow Erd\H{o}s-S\'os Conjectures

TL;DR

This paper explores the maximum edges in properly edge-colored hypercubes avoiding rainbow trees, proposing a rainbow extremal number variant and verifying an Erdős-Sós type conjecture for certain trees.

Contribution

It introduces the relative rainbow extremal number in hypercubes and verifies an Erdős-Sós type conjecture for specific families of trees.

Findings

Proposes the relative rainbow extremal number in hypercubes.

Verifies the Erdős-Sós type conjecture for some infinite tree families.

Provides insights into rainbow extremal problems in high-dimensional graphs.

Abstract

An edge colored graph is said to contain rainbow-$F$ if $F$ is a subgraph and every edge receives a different color. In 2007, Keevash, Mubayi, Sudakov, and Verstra\"ete introduced the \emph{rainbow extremal number} $\mathrm{ex}^*(n,F)$, a variant on the classical Tur\'an problem, asking for the maximum number of edges in a $n$-vertex properly edge-colored graph which does not contain a rainbow-$F$. In the following years many authors have studied the asymptotic behavior of $\mathrm{ex}^*(n,F)$ when $F$ is bipartite. In the particular case that $F$ is a tree $T$, the infamous Erd\"os-S\'os conjecture says that the extremal number of $T$ depends only on the size of $T$ and not its structure. After observing that such a pattern cannot hold for $\mathrm{ex}^*$ in the usual setting, we propose that the relative rainbow extremal number $\mathrm{ex}^*(Q_n,T)$ in the $n$-dimensional hypercube $Q_n$ will satisfy an Erd\"os-S\'os type Conjecture and verify it for some infinite families of trees $T$.