Fractional Colouring and Hadwigerʹs Conjecture

LetGbe a loopless graph with noKp+1minor. We prove that the “fractional chromatic number” ofGis at most 2p; that is, it is possible to assign a rationalq(S)⩾0 to every stable setS⊆V(G) so that ∑S∋v q(S)=1 for every vertexv, and ∑S q(S)⩽2p.