Given graphs G, H1 and H2, let G→mr(H1, H2) denote the property that in every edge-colouring of G there is a monochromatic copy of H1 or a rainbow copy of H2. The constrained Ramsey number, defined as the minimum n such that Kn→mr(H1, H2), exists if and only if H1 is a star or H2 is a forest. We determine the threshold for the property G(n,p) →mr(H1, H2) when H2 is a forest.

In this work, we study hitting times for the appearance of a spanning structure in the Erdős-Rényi random directed graph processes. Namely, we are concerned with the appearance of an arborescence, a spanning digraph in which, for a vertex u called the root and any other vertex v, there is exactly one directed path from u to v. Let D(n, 0), D(n, 1),..., D(n, n(n - 1)) be the random digraph process where for every m ∈ {0,..., n(n - 1)}, D(n, m) is a digraph with vertex set {1,...,n}; D(n, 0) has no arcs and, for 1 ≤ m ≤ n(n - 1), the digraph D(n,m) is obtained by adding an arc to D(n,m - 1), chosen uniformly at random among the not present arcs. In this paper we determine the hitting time for the existence of k arc-disjoint arborescences when k = k(n) ⩽ log n.