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Ramsey numbers of trees versus odd cycles
Burr, Erd\H{o}s, Faudree, Rousseau and Schelp initiated the study of Ramsey
numbers of trees versus odd cycles, proving that for all
odd and , where is a tree with vertices
and is an odd cycle of length . They proposed to study the minimum
positive integer such that this result holds for all ,
as a function of . In this paper, we show that is at most linear.
In particular, we prove that for all odd and
. Combining this with a result of Faudree, Lawrence, Parsons and
Schelp yields is bounded between two linear functions, thus
identifying up to a constant factor.Comment: 10 pages, updated to match EJC versio