Linear connectivity forces large complete bipartite minors

Let a be an integer. It is proved that for any s and k, there exists a constant N=N(s,k,a) such that every -connected graph with at least N vertices either contains a subdivision of Ka,sk or a minor isomorphic to s disjoint copies of Ka,k. In fact, we prove that connectivity 3a+2 and minimum degree at least are enough while the other conditions cannot be weakened.When s=1 and k=a, this implies that every -connected graph with at least N(a) vertices has a Ka-minor. This is the first result where a linear lower bound on the connectivity in terms of the parameter a forces a Ka-minor. This resolves a conjecture proposed by Mader [W. Mader, Existenz n-fach zusammenhängender Teilgraphen in Graphen genügend grosser Kantendichte, Abh. Math. Sem. Univ. Hamburg 37 (1972) 86–97] and Thomason [A. Thomason, An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984) 261–265; A. Thomason, The extremal function for complete minors, J. Combin. Theory Ser. B 81 (2001) 318–338]. Our result also generalizes a recent result of Böhme and Kostochka [T. Böhme, A. Kostochka, Disjoint Kr-minors in large graphs with given average degree, European J. Combin. 26 (2005) 289–292], resolves a conjecture of Fon-Der-Flaass [D. Fon-Der-Flaass, private communication], and has several other consequences.