Dense graphs have K_3,t minors

Let K_3,t^* denote the graph obtained from K _3,t by adding all edges between the three vertices of degree t in it. We prove that for each t<6300 and n<t+3, each n-vertex graph G with e(G)>12(t+3)(n-2)+1 has a K_3,t^*-minor. The bound is sharp in the sense that for every t, there are infinitely many graphs G with e(G)=12(t+3)(|V(G)|-2)+1 that have no K3,t-minor. The result confirms a partial case of the conjecture by Woodall and Seymour that every (s+t)-chromatic graph has a K_s,t-minor.