Degree conditions for Ramsey goodness of paths

A classical result of Chv\'atal implies that if n (r-1) (t-1) +1, then any colouring of the edges of Kₙ in red and blue contains either a monochromatic red Kᵣ or a monochromatic blue Pₜ. We study a natural generalization of his result, determining the exact minimum degree condition for a graph G on n = (r - 1) (t - 1) + 1 vertices which guarantees that the same Ramsey property holds in G. In particular, using a slight generalization of a result of Haxell, we show that (G) n - t/2 suffices, and that this bound is best possible. We also use a classical result of Bollob\'as, Erdos, and Straus to prove a tight minimum degree condition in the case r = 3 for all n 2t - 1.