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mean that if the edges of F are colored red and blue (say) in any fashion, then either the red subgraph of F contains a copy of G or the blue subgraph contains H. Write F-G for F } (G,G). A natural question to consider is that of characterizing those F for which F 3 (G,H) for a given G and H. This question is in general extremely difficult, although in a(More)
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)
It is shown that the classical Ramsey numbers T(m, ta) satisfy r(m,n) 2: r(m,n-1) + 2m-3, and.forl <k<n-2. r(m,n) >r(m,n-k)+r(m,k+ 1)-l. Consequences of the first result for some generalized Ramsey numbers will be considered. If m and n are integers 2 2, define the (classical) Ramsey number r(m, n) to be at least integer t such that if the edges of the(More)
A connected graph G of order n is called m-good if r(Km ,G) = (m-1)(n-1) + 1. Let f(m,n) be the largest integer q such that every connected graph of order n and size q is m-good and let g(m,n) be the largest q for which there exists a connected graph G of order n and size q which is m-good. Asymptotic bounds are given for f and g. 1. Tntroduction. One of(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)
ABSTRACr. Let G be a connected graph on n vertices with no more than n(1 + e) edges, and Pk or Ck a path or cycle with k vertices. In this paper we will show that if n is sufficiently large and a is sufficiently small then for k odd r(G, Ck) = 2n-1. Also, for k > 2, r(G,Pk)=max{n+[k/2]-1,n+k-2-á-ő}, where á is the independence number of an appropriate(More)