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 < e < 10-16 there exists a number t0 = t0(e) such that for all integers t ≥ t0 and s ≤ e7t/log t every graph of average degree at least (1 + e)t contains a Ks + K-t minor (and thus also a Ks,t minor). The bounds are essentially the best possible.