Proof of a conjecture on fractional Ramsey numbers

Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r f ( a 1 , a 2 , …, a k ) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k =2. In this article, we answer an open problem by determining an explicit formula for the general case k >2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 164–178, 2010