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Let G be a graph with chromatic number x a ° d with t being the minimum number of points in any color class of any point-coloring of G with x colors. Let H be any connected graph and let H n be a graph on n points which is homeomorphic to H. It is proved that if n is large enough, the Ramsey number r{G, H n) satisfies r(G, H n) = (x — l)(n —l) + r. It is… (More)

Chvátal has shown that if T is a tree on n points then r(Kk , T) _ (k-1) (n-1) + 1 , where r is the (generalized) Ramsey number. It is shown that the same result holds when T is replaced by many other graphs. Such a T is called k-good. The results proved all support the conjecture that any large graph that is sufficiently sparse, in the appropriate sense,… (More)

It is shown in this paper that the pair (G, H) is Ramsey infinite when both G and H are forests, with at least one of G or H having a non-star component. In addition, an infinite subfamily of R(PP.) is constructed .

It is shown that if G and H are star-forests with no single edge stars, then (G, H) is Ramsey-finite if and only if both G and H are single stars with an odd number of edges. Ramsey-finite or Ramsey-infinite depending on the choice of G, H, k, and l with the general case unsettled. This disproves the conjecture given in [2] where it is suggested that the… (More)