Chromatic Ramsey numbers and two-color Turán densities

Abstract: Given a graph $G$, its $2$-color Tur\'{a}n number $\mathrm{ex}^{(2)}(n,G)$ is the largest number of edges in an $n$-vertex graph whose edges can be colored with two colors avoiding a monochromatic copy of $G$. Let $\pi^{(2)}(G)=\lim_{n\to\infty}\mathrm{ex}^{(2)}(n,G)/\binom{n}{2}$ be the $2$-color Tur\'{a}n density of $G$. What real numbers in the interval $(0,1)$ are realized as the $2$-color Tur\'{a}n density of some graph? It is known that $\pi^{(2)}(G)=1-(R_{\chi}(G)-1)^{-1}$, where $R_{\chi}(G)$ is the chromatic Ramsey number of $G$. However, determining specific values of $R_{\chi}(G)$ is challenging. Burr, Erd\H{o}s, and Lov\'{a}sz showed that $(k-1)^2+1\leqslant{R_{\chi}(G)}\leqslant{R(k)}$, for any $k$-chromatic graph $G$, where $R(k)$ is the classical Ramsey number. The upper bound here can be attained by a clique and the lower bound is achieved by a graph constructed by Zhu. To the best of our knowledge, there are no other, besides these two, known values of $R_{\chi}(G)$ among $k$-chromatic graphs $G$ for general $k$. In this paper we prove that there are $\Omega(k)$ different values of $R_{\chi}(G)$ among $k$-chromatic graphs $G$. In addition, we determine a new value for the chromatic Ramsey numbers of $4$-chromatic graphs. This sheds more light into the possible $2$-color Tur\'{a}n densities of graphs.