Abstract
We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-\(2\) outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on \(2n\) vertices are bounded by \(O(n^{3})\) and \(O(n^{10})\), in the convex and general case, respectively. We then apply similar methods to prove an \(n^{O(\log (n))}\) upper bound on the Ramsey number of a path with \(n\) ordered vertices.
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Acknowledgments
This research was started at the 2nd Emléktábla Workshop held in Gyöngyöstarján, January 24–27, 2011. Research was supported by the project CE-ITI (GAČR P202/12/G061) of the Czech Science Foundation and by the Grant SVV-2014-260103. Josef Cibulka and Pavel Valtr were also supported by the project no. 52410 of the Grant Agency of Charles University. Pu Gao was supported by the Humboldt Foundation and is currently affiliated with University of Toronto. Marek Krčál was supported by the ERC Advanced Grant No. 267165. The authors would like to thank to Gyula Károlyi for introduction to the geometric Ramsey theory and to Jan Kynčl and Martin Balko for discussions about the Ramsey theory of ordered graphs. The authors are grateful to the anonymous referees for their valuable comments.
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An extended abstract of this paper appeared in the proceedings of the EuroComb 2013 conference.
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Cibulka, J., Gao, P., Krčál, M. et al. On the Geometric Ramsey Number of Outerplanar Graphs. Discrete Comput Geom 53, 64–79 (2015). https://doi.org/10.1007/s00454-014-9646-x
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DOI: https://doi.org/10.1007/s00454-014-9646-x