Pach, JánosIST Austria; Tomon, István
An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer k, there exists a constant ck > 0 such that any ordered graph G on n vertices with the property that neither G nor its complement contains an induced monotone path of size k, has either a clique or an independent set of size at least n^ck . This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs. A key idea of the above paper was to show that any unordered graph on n vertices that does not contain an induced path of size k, and whose maximum degree is at most c(k)n for some small c(k) > 0, contains two disjoint linear size subsets with no edge between them. This approach fails for ordered graphs, because the analogous statement is false for k ≥ 3, by a construction of Fox. We provide some further examples showing that this statement also fails for ordered graphs avoiding other ordered trees.
Journal of Combinatorial Theory. Series B
We would like to thank the anonymous referees for their useful comments and suggestions. János Pach is partially supported by Austrian Science Fund (FWF) grant Z 342-N31 and by ERC Advanced grant “GeoScape.” István Tomon is partially supported by Swiss National Science Foundation grant no. 200021_196965, and thanks the support of MIPT Moscow. Both authors are partially supported by The Russian Government in the framework of MegaGrant no. 075-15-2019-1926.
Pach J, Tomon I. Erdős-Hajnal-type results for monotone paths. Journal of Combinatorial Theory Series B. 2021;151:21-37. doi:10.1016/j.jctb.2021.05.004
Pach, J., & Tomon, I. (2021). Erdős-Hajnal-type results for monotone paths. Journal of Combinatorial Theory. Series B. Elsevier. https://doi.org/10.1016/j.jctb.2021.05.004
Pach, János, and István Tomon. “Erdős-Hajnal-Type Results for Monotone Paths.” Journal of Combinatorial Theory. Series B. Elsevier, 2021. https://doi.org/10.1016/j.jctb.2021.05.004.
J. Pach and I. Tomon, “Erdős-Hajnal-type results for monotone paths,” Journal of Combinatorial Theory. Series B, vol. 151. Elsevier, pp. 21–37, 2021.
Pach J, Tomon I. 2021. Erdős-Hajnal-type results for monotone paths. Journal of Combinatorial Theory. Series B. 151, 21–37.
Pach, János, and István Tomon. “Erdős-Hajnal-Type Results for Monotone Paths.” Journal of Combinatorial Theory. Series B, vol. 151, Elsevier, 2021, pp. 21–37, doi:10.1016/j.jctb.2021.05.004.
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