A generalization of Ramsey theory for linear forests

Chung and Liu defined the d-chromatic Ramsey numbers as a generalization of Ramsey numbers by replacing the usual condition with a slightly weaker condition. Let 1 < d < c and let t = (c d). Assume A(1), A(2), ..., A(t) are all d-subsets of a set containing c distinct colours. Let G(1), G(2), ..., G(t) be graphs. The d-chromatic Ramsey number denoted by r(d)(c) (G(1), G(2), ..., G(t)) is defined as the least number p such that, if the edges of the complete graph K-p are coloured in any fashion with c colours, then for some i, the subgraph whose edges are coloured by colours in A(i) contains a G(i). In this paper, we determine r(t-1)(t) (G(1), G(2), ..., G(t)) for t = 3, 4 and for linear forests G(i), 1 = i = t, when G(1) has at most one odd component and G(j), 2 <= j <= t - 1, either is a path or has no odd component. Consequently, these numbers are determined when G(i), 1 <= i <= t, is either a path or a stripe.