Proof of a conjecture on fractional Ramsey numbers

: Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function rf (a1,a2, ...,ak) 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

• Type: Journal
• Year: 2010
• Where: JGT
• Authors: Jason Brown, Richard Hoshino
• Keywords: Fractional Ramsey Function, JGT 2010, Ramsey Function Rf, Star Extremal Graphs