Large complete minors in random subgraphs

Let G be a graph of minimum degree at least k and let G p be the random subgraph of G obtained by keeping each edge independently with probability p . We are interested in the size of the largest complete minor that G p contains when p = (1 + ε )/ k with ε > 0. We show that with high probability G p contains a complete minor of order $\tilde{\Omega}(\sqrt{k})$ , where the ~ hides a polylogarithmic factor. Furthermore, in the case where the order of G is also bounded above by a constant multiple of k , we show that this polylogarithmic term can be removed, giving a tight bound.