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An ordered graph is a graph with a linear ordering on its vertices. The online Ramsey game for ordered graphs $$G$$ and $$H$$ is played on an infinite sequence of vertices; on each turn, Builder draws an edge between two vertices, and Painter colors it red or blue. Builder tries to create a red $$G$$ or a blue $$H$$ as quickly as possible, while Painter wants the opposite. The online ordered Ramsey number $$r_o(G,H)$$ is the number of turns the game lasts with optimal play. In this paper, we consider the behavior of $$r_o(G,P_n)$$ for fixed $$G$$, where $$P_n$$ is the monotone ordered path. We prove an $$O(n \log n)$$ bound on $$r_o(G,P_n)$$ for all $$G$$ and an $O(n)$ bound when $$G$$ is $$3$$-ichromatic; we partially classify graphs $$G$$ with $$r_o(G,P_n) = n + O(1)$$. Many of these results extend to $$r_o(G,C_n)$$, where $$C_n$$ is an ordered cycle obtained from $$P_n$$ by adding one edge.
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On edge‐ordered Ramsey numbers
An edge‐ordered graph is a graph with a linear ordering of its edges. Two edge‐ordered graphs areequivalentif there is an isomorphism between them preserving the edge‐ordering. Theedge‐ordered Ramsey number redge(H; q) of an edge‐ordered graphHis the smallestNsuch that there exists an edge‐ordered graphGonNvertices such that, for everyq‐coloring of the edges ofG, there is a monochromatic subgraph ofGequivalent toH. Recently, Balko and Vizer proved thatredge(H; q) exists, but their proof gave enormous upper bounds on these numbers. We give a new proof with a much better bound, showing there exists a constantcsuch thatfor every edge‐ordered graphHonnvertices. We also prove a polynomial bound for the edge‐ordered Ramsey number of graphs of bounded degeneracy. Finally, we prove a strengthening for graphs where every edge has a label and the labels do not necessarily have an ordering.
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- Award ID(s):
- 1855635
- PAR ID:
- 10260102
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 57
- Issue:
- 4
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- p. 1174-1204
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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