Complete Minors In K _s,s -Free Graphs

We prove that for a fixed integer s≥2 every K _s,s -free graph of average degree at least r contains a K _p minor where $p = r^{{1 + \frac{1} {{2{\left( {a - 1} \right)}}} + o{\left( 1 \right)}}}$. A well-known conjecture on the existence of dense K _s,s -free graphs would imply that the value of the exponent is best possible. Our result implies Hadwiger’s conjecture for K _s,s -free graphs whose chromatic number is sufficiently large compared with s.