On the edit distance function of the random graph

Given a hereditary property of graphs H and a p is an element of [0, 1], the edit distance function ed(H)(p) is asymptotically the maximum proportion of edge additions plus edge deletions applied to a graph of edge density p sufficient to ensure that the resulting graph satisfies H. The edit distance function is directly related to other well-studied quantities such as the speed function for H and the H-chromatic number of a random graph. Let H be the property of forbidding an Erdos-Renyi random graph F similar to G(n(0), P-0), and let phi represent the golden ratio. In this paper, we show that if p(0) is an element of [1 - 1/phi, 1/phi], then a.a.s. as n(0) -> infinity. ed(H)(p) = (1 + o(1)) 2 log n(0)/n(0).min {p/-log(1-P-0), 1-p/-log p(0)}. Moreover, this holds for p is an element of [1/3, 2/3] for any P-0 is an element of (0, 1). A primary tool in the proof is the categorization of p-core coloured regularity graphs in the range p is an element of [1 - 1/phi, 1/phi]. Such coloured regularity graphs must have the property that the non-grey edges form vertex-disjoint cliques.