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- W1670697891 abstract "Abstract In this article we prove a new result about partitioning colored complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for , in every edge coloring of with the colors red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete ‐partite graph. When the coloring of is connected in red, we prove a stronger result—that it is possible to cover all the vertices with k red paths and a blue balanced complete ‐partite graph. Using these results we determine the Ramsey number of an n ‐vertex path, , versus a balanced complete t ‐partite graph on vertices, , whenever . We show that in this case , generalizing a result of Erdős who proved the case of this result. We also determine the Ramsey number of a path versus the power of a path . We show that , solving a conjecture of Allen, Brightwell, and Skokan." @default.
- W1670697891 created "2016-06-24" @default.
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- W1670697891 date "2016-04-25" @default.
- W1670697891 modified "2023-10-05" @default.
- W1670697891 title "Calculating Ramsey Numbers by Partitioning Colored Graphs" @default.
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- W1670697891 doi "https://doi.org/10.1002/jgt.22036" @default.
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