Forcing unbalanced complete bipartite minors

Myers conjectured that for every integer s there exists a positive constant C such that for all integers t every graph of average degree at least Ct contains a Ks,t minor. We prove the following stronger result: for every 0<ε<10−16 there exists a number t0=t0(ε) such that for all integers t≥t0 and s≤ε7t/logt every graph of average degree at least (1+ε)t contains a Ks+Kt minor (and thus also a Ks,t minor). The bounds are essentially the best possible.