Online Ramsey numbers of ordered paths and cycles

Abstract

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 $O(n \log_2n)$ 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.

Figures (15)

• Figure 1: The three two-edge graphs of Theorem \ref{['thm:bad-small-graphs']}
• Figure 2: Examples of the St. Ives matching and the partial St. Ives matching
• Figure 3: Builder's strategy for Theorem \ref{['thm:left-degree']}, where $G = K_4$. The vertical positions of vertices are only varied for clarity; in fact, the vertices on different blue paths or in different $G_i$ are not required to be in any particular order relative to each other, or even to be distinct.
• Figure 4: An illustration of the definition of the sets $A,B,$ and $C$ where $a=b=2$ and $c=d=1$.
• Figure 5: An example of left-forced and right-forced edges in the proof of Lemma \ref{['lemma:p3-lower-bound']}

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.5
• Theorem 1.6
• proof : Proof of Theorem \ref{['thm:left-degree']}
• Claim 2.1