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On the size-Ramsey number of grid graphs

Part of: Extremal combinatorics

Published online by Cambridge University Press:  06 April 2021

Dennis Clemens ,
Meysam Miralaei ,
Damian Reding ,
Mathias Schacht  and
Anusch Taraz
Dennis Clemens
Affiliation:
Institut für Mathematik, TU Hamburg, Hamburg, Germany
Meysam Miralaei
Affiliation:
Department of Mathematics, Isfahan University of Technology, Isfahan, Iran
Damian Reding
Affiliation:
Institut für Mathematik, TU Hamburg, Hamburg, Germany
Mathias Schacht*
Affiliation:
Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany
Anusch Taraz
Affiliation:
Institut für Mathematik, TU Hamburg, Hamburg, Germany
*
*Corresponding author. Email: [email protected]
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Abstract

The size-Ramsey number of a graph F is the smallest number of edges in a graph G with the Ramsey property for F, that is, with the property that any 2-colouring of the edges of G contains a monochromatic copy of F. We prove that the size-Ramsey number of the grid graph on n × n vertices is bounded from above by n3+o(1).

MSC classification

Primary: 05D10: Ramsey theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 5 , September 2021 , pp. 670 - 685
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

M. Miralaei was supported by the Ministry of Science, Research and Technology of Iran and part of the research was carried out during a visit to the University of Hamburg.

‡

M. Schacht was partly supported by the European Research Council (PEPCo 724903).

References

Allen, P., Böttcher, J., Hàn, H., Kohayakawa, Y. and Person, Y. Blow-up lemmas for sparse graphs. Submitted.Google Scholar
Alon, N. and Chung, F. R. K. (1988) Explicit construction of linear sized tolerant networks. Discrete Math. 72 15–19.CrossRefGoogle Scholar
Beck, J. (1983) On size Ramsey number of paths, trees, and circuits, I. J. Graph Theory 7 115–129.CrossRefGoogle Scholar
Beck, J. (1990) On size Ramsey number of paths, trees and circuits, II. In Mathematics of Ramsey Theory (Nešetřil, J. and Rödl, V., eds), pp. 34–45. Springer.CrossRefGoogle Scholar
Bollobás, B. (2001) Random Graphs, Vol. 73 of Cambridge Studies in Advanced Mathematics, second edition. Cambridge University Press.CrossRefGoogle Scholar
Clemens, D., Jenssen, M., Kohayakawa, Y., Morrison, N., Mota, G. O., Reding, D. and Roberts, B. (2019) The size-Ramsey number of powers of paths. J. Graph Theory 91 290–299.CrossRefGoogle Scholar
Conlon, D. and Nenadov, R. Size Ramsey numbers of triangle-free graphs with bounded degree. Preprint.Google Scholar
Dellamonica, D., Jr (2012) The size-Ramsey number of trees. Random Struct. Algorithms 40 49–73.CrossRefGoogle Scholar
Dudek, A. and Prałat, P. (2015) An alternative proof of the linearity of the size-Ramsey number of paths. Combin. Probab. Comput. 24 551–555.CrossRefGoogle Scholar
Dudek, A. and Prałat, P. (2017) On some multicolor Ramsey properties of random graphs. SIAM J. Discrete Math. 31 2079–2092.CrossRefGoogle Scholar
Erdős, P. (1981) On the combinatorial problems which I would most like to see solved. Combinatorica 1 25–42.CrossRefGoogle Scholar
Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H. (1978) The size Ramsey number. Period. Math. Hungar. 9 145–161.CrossRefGoogle Scholar
Friedman, J. and Pippenger, N. (1987) Expanding graphs contain all small trees. Combinatorica 7 71–76.CrossRefGoogle Scholar
Gerke, S., Kohayakawa, Y., Rödl, V. and Steger, A. (2007) Small subsets inherit sparse ε-regularity. J. Combin. Theory Ser. B 97 34–56.CrossRefGoogle Scholar
Haxell, P. E. and Kohayakawa, Y. (1995) The size-Ramsey number of trees. Israel J. Math. 89 261–274.CrossRefGoogle Scholar
Haxell, P. E., Kohayakawa, Y. and Łuczak, T. (1995) The induced size-Ramsey number of cycles. Combin. Probab. Comput. 4 217–239.CrossRefGoogle Scholar
Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience.CrossRefGoogle Scholar
Javadi, R., Khoeini, F., Omidi, G. R. and Pokrovskiy, A. (2019) On the size-Ramsey number of cycles. Combin. Probab. Comput. 28 871–880.CrossRefGoogle Scholar
Kohayakawa, Y. (1997) Szemerédi’s regularity lemma for sparse graphs. In Foundations of Computational Mathematics (Cucker, F. and Shub, M., eds), pp. 216–230. Springer.CrossRefGoogle Scholar
Kohayakawa, Y. and Rödl, V. (2003) Regular pairs in sparse random graphs, I. Random Struct. Algorithms 22 359–434.CrossRefGoogle Scholar
Kohayakawa, Y., Rödl, V., Schacht, M. and Szemerédi, E. (2011) Sparse partition universal graphs for graphs of bounded degree. Adv. Math. 226 5041–5065.CrossRefGoogle Scholar
Letzter, S. (2016) Path Ramsey number for random graphs. Combin. Probab. Comput. 25 612–622.CrossRefGoogle Scholar
Rödl, V. and Duke, R. A. (1985) On graphs with small subgraphs of large chromatic number. Graphs Combin. 1 91–96.CrossRefGoogle Scholar
Rödl, V. and Szemerédi, E. (2000) On size Ramsey numbers of graphs with bounded degree. Combinatorica 20 257–262.Google Scholar
Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes, Vol. 260 of Colloq. Internat. CNRS, pp. 399–401. CNRS.Google Scholar