Constrained colourings of random graphs

Given graphs G, H-1 and H-2, let G (sic) (H-1, H-2) denote the property that in every edge -colouring of G there is a monochromatic copy of H-1 or a rainbow copy of H-2. The constrained Ramsey number, defined as the minimum n such that K-n (sic) (H-1, H-2), exists if and only if H-1 is a star or H-2 is a forest. We determine the threshold for the property G(n, p) (sic) (H-1, H-2) when H-2 is a forest. (C) 2021 The Authors. Published by Elsevier B.V.