On the ultimate lexicographic Hall-ratio

The Hall-ratio ρ ( G ) of a graph G is the ratio of the number of vertices and the independence number maximized over all subgraphs of G . The ultimate lexicographic Hall-ratio of a graph G is defined as lim n → ∞ ρ ( G ∘ n ) n , where G ∘ n denotes the n th lexicographic power of G (that is, n times repeated substitution of G into itself). Here we prove the conjecture of Simonyi stating that the ultimate lexicographic Hall-ratio equals the fractional chromatic number for all graphs.