Xuding Zhu: Fractional Hedetniemi's conjecture, Chromatic Ramsey number and circular chromatic Ramsey number.
For graphs H,F,G, we write H⟶(F,G) to mean that whenever the edges of H are coloured by red and blue, there is either red copy of F or a blue copy of G. Let ρ be a monotone graph parameter. The ρ-Ramsey number of (F,G), written as Rρ(F,G), is the infimum of ρ(H) such that H⟶(G,H). We write Rρ(G) for Rρ(G,G). We are interested in the case that ρ is the chromatic number and circular chromatic number. Two particular questions are the following:
Given an integer n, determine min{Rχ(G): χ(G)=n}.
Given a rational number r, determine inf{Rχc(G): χc(G)=r}.
These problems are related to the chromatic number, circular chromatic number and fractional chromatic number of the product of graphs. In this talk, I shall sketch a proof of the fractional version of Hedetniemi's conjecture: χf(G×H) = min {χf(G), χf(H)}. Then explain how it is used in the study of the above two problems.
