Clique minors in graphs with a forbidden subgraph

The classical Hadwiger conjecture dating back to 1940s states that any graph of chromatic number at least r has the clique order as a minor. Hadwiger's is an example well-studied class problems asking how large minor one can guarantee in with certain restrictions. One problem this type asks what largest size on n vertices independence most r. If true would imply existence . Results Kühn and Osthus Krivelevich Sudakov if assumes addition G H-free for some bipartite H then find polynomially larger This recently been extended triangle-free graphs by Dvořák Yepremyan, answering question Norin. We complete picture show same arbitrary H, Yepremyan. In particular, we -free , constant depending only s. exponent result tight up factor front term.