Chromatic Ramsey Numbers and Two‐Color Turán Densities

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