Highly connected subgraphs of graphs with given independence number

Abstract Let G be a graph on n vertices with independence number α . We prove that if n is sufficiently large ( n ≥ α 2 k + 1 will do), then G always contains a k -connected subgraph on at least n ∕ α vertices. The value of n ∕ α is sharp, since G might be the disjoint union of α equally-sized cliques. For k ≥ 3 and α = 2 , 3 , we shall prove that the same result holds for n ≥ 4 ( k − 1 ) and n ≥ 27 ( k − 1 ) 4 respectively, and that these lower bounds on n are sharp.