Many disjoint dense subgraphs versus large k-connected subgraphs in large graphs with given edge density

It is proved that for all positive integers d, k, s, t with t ≥ k + 1 there is a positive integer M = M (d, k, s, t) such that every graph with edge density at least d + k and at least M vertices contains a k-connected subgraph on at least t vertices, or s pairwise disjoint subgraphs with edge density at least d. By a classical result of Mader [W. Mader, Existenz n-fach zusammenhängender Teilgraphen in Graphen genügend großer Kantendichte, Abh. Math. Sem Univ. Hamburg, 37 (1972) 86-97] this implies that every graph with edge density at least 3 k and sufficiently many vertices contains a k-connected subgraph with at least r vertices, or r pairwise disjoint k-connected subgraphs. Another classical result of Mader [W. Mader, Homomorphiesätze für Graphen, Math. Ann. 178 (1968) 154-168] states that for every n there is an l (n) such that every graph with edge density at least l (n) contains a minor isomorphic to K_n. Recently, it was proved in [T. Böhme, K. Kawarabayashi, J. Maharry, B. Mohar, Linear connectivity forces dense minors, J. Combin. Theory Ser. B (submitted for publication)] that every (frac(31, 2) a + 1)-connected graph with sufficiently many vertices either has a topological minor isomorphic to K_{a, p q}, or it has a minor isomorphic to the disjoint union of p copies of K_{a, q}. Combining these results with the result of the present note shows that every graph with edge density at least l (a) + (frac(31, 2) a + 1) and sufficiently many vertices has a topological minor isomorphic to K_{a, p a}, or a minor isomorphic to the disjoint union of p copies of K_a. This implies an affirmative answer to a question of Fon-der-Flaass.