Abstract
We 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 also prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases.
This research was started at the 2nd Emléktabla 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-2012-265313 (Discrete Methods and Algorithms). 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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B. Aronov, P. Erdős, W. Goddard, D. J. Kleitman, M. K$lugerman, J. Pach and L. J. Schulman, Crossing families, Combinatorica 14(2) (1994), 127–134.
G. T. Chen and R. H. Schelp, Graphs with linearly bounded Ramsey numbers, Journal of Combinatorial Theory, Series B 57(1) (1993), 138–149.
P. Gritzmann, B. Mohar, J. Pach and R. Pollack, Embedding a planar triangulation with vertices at specified points, Am. Math. Monthly 98 (1991), 165–166 (Solution to problem E3341).
G. Károlyi, J. Pach and G. Tóth, Ramsey-type results for geometric graphs, I, Discrete & Computational Geometry 18(3) (1997), 247–255.
G. Károlyi, J. Pach, G. Tóth and P. Valtr, Ramsey-type results for geometric graphs, II, Discrete & Computational Geometry 20(3) (1998), 375–388.
N. Robertson and P. D. Seymour, Graph minors. I. Excluding a forest, Journal of Combinatorial Theory, Series B 35(1) (1983), 39–61.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2013 Scuola Normale Superiore Pisa
About this paper
Cite this paper
Cibulka, J., Gao, P., Krčál, M., Valla, T., Valtr, P. (2013). Polynomial bounds on geometric Ramsey numbers of ladder graphs. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_28
Download citation
DOI: https://doi.org/10.1007/978-88-7642-475-5_28
Publisher Name: Edizioni della Normale, Pisa
Print ISBN: 978-88-7642-474-8
Online ISBN: 978-88-7642-475-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)