Rainbow Erdős-Sós Conjectures

TL;DR

This work studies rainbow extremal numbers in a relative host framework, focusing on the hypercube $Q_n$ as a natural host for rainbow analogues of the Erdős-Sós conjecture. It defines ex^*(G,F) and related invariants, and develops minimum-degree embedding techniques to enforce rainbow embeddings of trees. The authors establish exact values for $ex^*(Q_n,P_3)$ and $ex^*(Q_n,P_4)$ and identify broad families of trees (e.g., paths, pendant-paths with many leaves, stars, certain spiders) for which $ex^*(Q_n,T)=(k-1)/2\,2^n$, along with a general upper bound of the form $ex^*(G,T)<(2k-1)|V(G)|$. These results support a Rainbow Erdős-Sós-type conjecture on $Q_n$ and illustrate how hypercube structure governs rainbow-avoidance, with implications for broader host graphs and tree families.

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 Verstraëte introduced the \emph{rainbow extremal number} $\mathrm{ex}^*(n,F)$, a variant on the classical Turán 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 Erdös-Sós 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 Erdös-Sós type Conjecture and verify it for some infinite families of trees $T$.

Figures (8)

• Figure 1: The four problems considered here for a $k$-edge tree $T$
• Figure 2:
• Figure 3:
• Figure 4:
• Figure 5: From top to bottom: 1) $r,s\leq m$ 2) $m <r,s$ 3) $r\leq m < s$ with $r \leq t-s$ 4) $r\leq m < s$ with $r>t-s$

Theorems & Definitions (28)

• Conjecture 1: Erdös-Sós
• Remark 1
• Proposition 1
• Conjecture 2
• Conjecture 3
• Theorem 2
• Theorem 3
• Proposition 4
• proof : Proof of Proposition \ref{['prop:delta_n']} Part \ref{['item:delta_n_ub']}
• proof : Proof of Proposition \ref{['prop:delta_n']} Part \ref{['item:delta_n_star']}