Online Ramsey numbers of the claw versus cycles

TL;DR

This work analyzes the online Ramsey numbers $\tilde r(G,H)$ in a Builder–Painter game, focusing on $G=K_{1,3}$ and long cycles $H=C_\ell$. It achieves an exact value $\tilde r(K_{1,3},C_\ell)=\left\lfloor \frac{3(\ell+1)}{2} \right\rfloor$ for all $\ell\ge 13$ by combining a Painter-friendly lower bound with a constructive Builder strategy. The upper-bound strategy introduces novel concepts—good/better/best vertices, blue main paths, Type 1/2/3 configurations, and the Blue Path Expansion Algorithm—together with the wavy path construct to extend blue paths to a target length before closing a blue cycle, all controlled by a suite of lemmas. This advances the understanding of online Ramsey behavior for sparse graphs and long cycles, providing exact results and a toolkit for handling similar online Ramsey scenarios.

Abstract

The online Ramsey number $\tilde r(G,H)$ is defined via a Builder--Painter game on an empty graph with countably many vertices. In each round, Builder reveals an edge, which Painter immediately colors either red or blue. Builder wins once a red copy of $G$ or a blue copy of $H$ appears, and $\tilde r(G,H)$ is the minimum number of edges Builder must reveal to force a win. For a long cycle $C_\ell$, the online Ramsey numbers $\tilde r(G,C_\ell)$ are known only for a few specific choices of $G$. In particular, exact values were determined for $G=P_3$ by Cyman, Dzido, Lapinskas, and Lo (Electron. J. Combin., 2015), while asymptotically tight results were obtained when $G$ is an even cycle by Adamski, Bednarska-Bzdȩga, and Blažej (SIAM J. Discrete Math., 2024). In this paper, we consider the case where $G$ is the claw $K_{1,3}$ and determine the exact value of $\tilde r(K_{1,3},C_\ell)$. We show that \[ \tilde r(K_{1,3},C_\ell)=\left\lfloor \frac{3(\ell+1)}{2} \right\rfloor \quad \text{for all } \ell \ge 13. \]

Figures (8)

• Figure 3-1:
• Figure 3-2: Wavy Path
• Figure 3-3: The two drawings of $P(4,1)$ and the drawing of $P(4,0)$.
• Figure 3-4:
• Figure 3-5:

Theorems & Definitions (23)

• Theorem 1.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• proof
• Theorem 3.7
• proof
• Theorem 3.8