Abstract
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 K s, t minor. We prove the following stronger result: for every 0< ε<10 −16 there exists a number t 0= t 0( ε) such that for all integers t≥ t 0 and s≤ ε 7 t/log t every graph of average degree at least (1+ ε) t contains a K s+ K t minor (and thus also a K s, t minor). The bounds are essentially the best possible.
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