The Shannon Capacity of a Union

For an undirected graph G = (V,E), let G denote the graph whose vertex set is V n in which two distinct vertices (u1, u2, . . . , un) and (v1, v2, . . . , vn) are adjacent iff for all i between 1 and n either ui = vi or uivi ∈ E. The Shannon capacity c(G) of G is the limit limn→∞(α(G)), where α(G) is the maximum size of an independent set of vertices in G. We show that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities. This disproves a conjecture of Shannon raised in 1956.