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 Ω^~ (^√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.