Chromatic Ramsey Numbers and Two-Color Turán Densities

Abstract

Given a graph $G$, its 2-color Turán number ${\text{ex}}^{\left(2\right)}\left(n,G\right)$ is the maximum number of edges in an $n$-vertex graph, such that the edges can be colored with two colors avoiding a monochromatic copy of $G$. Let ${\pi }^{\left(2\right)}\left(G\right)={\lim }_{n\to \infty }{\text{ex}}^{\left(2\right)}(n,G)/\left(\frac{n}{2}\right)$ be the 2-color Turán density of $G$. What real numbers in the interval $\left(0,1\right)$ are realized as the 2-color Turán density of some graph? It is known that ${\pi }^{\left(2\right)}\left(G\right)=1-{\left(R_{\chi }\left(G\right)-1\right)}^{-1}$, where $R_{\chi }\left(G\right)$ is the chromatic Ramsey number of $G$. Burr, Erdős, and Lovász showed that ${\left(k-1\right)}^2+1\le R_{\chi }\left(G\right)\le R\left(k\right)$, for any $k$-chromatic graph $G$, where $R\left(k\right)$ is the classical Ramsey number. However, it is an open problem to determine how many distinct values between ${\left(k-1\right)}^2+1$ and $R\left(k\right)$ can be realized as $R_{\chi }\left(G\right)$ of some $k$-chromatic graph $G$ for general $k$. In this paper, among others, we prove that there are $\Omega \left(k\right)$ different values of $R_{\chi }\left(G\right)$ among $k$-chromatic graphs $G$. This sheds more light on the possible 2-color Turán densities of graphs.