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Brown, Nowakowski and Rall defined the ultimate categorical independenceratio of a graph G as A(G)=\lim_{k\to \infty} i(G^{\times k}), wherei(G)=\frac{\alpha (G)}{|V(G)|} denotes the independence ratio of a graph G, andG^{\times k} is the k-th categorical power of G. Leta(G)=\max{\frac{|U|}{|U|+|N_G(U)|}: U is an independent set of G}}, whereN_G(U) is the neighborhood of U in G. In this paper we answer a question ofAlon and Lubetzky, namely we prove that if a(G)\le 1/2 then A(G)=a(G), and ifa(G)>1/2 then A(G)=1. We also discuss some other open problems related to A(G)which are immediately settled by this result.

Answer to a question of Alon and Lubetzky about the ultimate categorical independence ratio