On rainbow Turán Densities of Trees

Abstract

For a given collection $\mathcal{G} = (G_1,\dots, G_k)$ of graphs on a common vertex set $V$, which we call a \emph{graph system}, a graph $H$ on a vertex set $V(H) \subseteq V$ is called a \emph{rainbow subgraph} of $\mathcal{G}$ if there exists an injective function $ψ:E(H) \rightarrow [k]$ such that $e \in G_{ψ(e)}$ for each $e\in E(H)$. The maximum value of $\min_{i}\{|E(G_i)|\}$ over $n$-vertex graph systems $\mathcal{G}$ having no rainbow subgraph isomorphic to $H$ is called the rainbow Turán number $\mathrm{ex}_k^{\ast}(n, H)$ of $H$. In this article, we study the rainbow Turán density $π_k^{\ast}(T) = \lim_{n \rightarrow \infty} \frac{\mathrm{ex}_k^{\ast}(n, T)}{\binom{n}{2}}$ of a tree $T$. While the classical Turán density $π(H)$ of a graph $H$ lies in the set $\{1-\frac{1}{t} : t\in \mathbb{N}\}$, the rainbow Turán density exhibits different behaviors as it can even be an irrational number. Nevertheless, we conjecture that the rainbow Turán density is always an algebraic number. We provide evidence for this conjecture by proving that the rainbow Turán density of a tree is an algebraic number. To show this, we identify the structure of extremal graphs for rainbow trees. Moreover, we further determine all tuples $(α_1,\dots, α_k)$ such that every graph system $(G_1,\dots,G_k)$ satisfying $|E(G_i)|>(α_i+o(1))\binom{n}{2}$ contains all rainbow $k$-edge trees. In the course of proving these results, we also develop the theory on the limit of graph systems.