Abstract
We give some improved estimates for the digraph Ramsey numbersr(K * n ,L m ), the smallest numberp such that any digraph of orderp either has an independent set ofn vertices or contains a transitive tournament of orderm.
By results of Baumgartner and of Erdős and Rado, this is equivalent to the following infinite partition problem: for an infinite cardinal κ and positive integersn andm, find the smallest numberp such that
that is, find the smallest numberp so that any graph whose vertices are well ordered where order type κ·p either has an independent subset of order type κ·n or a complete subgraph of sizem.
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This work was partly supported by grant number DMS9306286 from the National Science Foundation.
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Larson, J.A., Mitchell, W.J. On a problem of Erdős and Rado. Annals of Combinatorics 1, 245–252 (1997). https://doi.org/10.1007/BF02558478
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DOI: https://doi.org/10.1007/BF02558478