K_s,t minors in (s + t) - Chromatic graphs, II

Let K_s,t* denote the graph obtained from the complete graph K_s+t by deleting the edges of some K_t-subgraph. The author proved earlier that for each fixed s and t>t₀(s):= max{4^15s2+s,(240slog₂s)^8slog2s+1}, every graph with chromatic number s+t has a K_s,t* minor. This confirmed a partial case of the corresponding conjecture by Woodall and Seymour. In this paper, we show that the statement holds already for much smaller t, namely, for t>C(slogs)³.