# 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 *R _{ρ}(F,G)*, is the infimum of

*ρ(H)*such that

*H⟶(G,H)*. We write

*R*for

_{ρ}(G)*R*. We are interested in the case that

_{ρ}(G,G)*ρ*is the chromatic number and circular chromatic number. Two particular questions are the following:

Given an integer *n*, determine *min{R _{χ}(G): χ(G)=n}*.

Given a rational number *r*, determine *inf{R _{χc}(G): χ_{c}(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:
*χ _{f}(G×H) = min {χ_{f}(G), χ_{f}(H)}*. Then explain how it
is used in the study of the above two problems.