Chromatic Ramsey Numbers and Two‐Color Turán Densities

Given a graph , its 2‐color Turán number is the maximum number of edges in an ‐vertex graph, such that the edges can be colored with two colors avoiding a monochromatic copy of . Let be the 2‐color Turán density of . What real numbers in the interval are realized as the 2‐color Turán density of some graph? It is known that , where is the chromatic Ramsey number of . Burr, Erdős, and Lovász showed that , for any ‐chromatic graph , where is the classical Ramsey number. However, it is an open problem to determine how many distinct values between and can be realized as of some ‐chromatic graph for general . In this paper, among others, we prove that there are different values of among ‐chromatic graphs . This sheds more light on the possible 2‐color Turán densities of graphs.