The Hall ratio of a graph G is the maximum of |V (H)|/\alpha(H) over all subgraphs H of G. It is easy to see that the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic number is upper bounded by a function of the Hall ratio, and this has been disproved. We prove a conjecture of Walczak that there are graphs with arbitrarily large fractional chromatic number and Hall ratio arbitrarily close to 2. The proof uses probabilistic construction together with structural analysis. This is best possible since odd cycles have Hall ratio greater than 2.

Joint work with James Davies and  Meike Hatzel.