R. H. Schelp conjectured that if G is a graph with |V (G)| = R(Pn, Pn) such that δ(G) > 3|V (G)| 4 , then in every 2-colouring of the edges of G there is a monochromatic Pn. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree. Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching–matching Ramsey number satisfying R(Sn, nK2, nK2) = 3n− 1. This extends R(nK2, nK2) = 3n− 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp’s conjecture in an asymptotic sense through a standard application of the Regularity Lemma.