Abstract
AbstractWe shall investigate the behavior of the Ramsey function r(L, Tn), where L is a fixed graph and Tn is a tree on n vertices, while n ← ∞. We prove that in this case the appropriate Ramsey problem can be reduced to a corresponding Turán type extremal problem: if we take a maximum N for which a KN can be colored in two colors so that the first color contains no L and the second color contains no Tn, then by the results of Erdös, Faudree, Rousseau and Schelp, N = (p ‐ 1)n + 0(n), where p is the chromatic number of L. We shall prove that in the corresponding Ramsey‐coloring the edges of the first color from an “almost Turán graph” on p ‐ 1 classes. We shall also describe how the finer structure of L influences this Ramsey number and the stability of the Ramsey coloring. The most important new results—compared with some earlier results—are that we can guarantee structural properties of the extremal colorings.
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