Sparse multipartite graphs as partition universal for graphs with bounded degree

For graphs G and H, let signify that any red/blue edge coloring of G contains a monochromatic H as a subgraph. Denote . For any and n, we say that G is partition universal for if for every . Let be the random spanning subgraph of the complete r-partite graph with N vertices in each part, in which each edge of appears with probability p independently and randomly. We prove that for fixed there exist constants r, B and C depending only on such that if and , then asymptotically almost surely is partition universal for H(Delta, n).