Proof of a Conjecture on Online Ramsey Numbers of Stars versus Paths

TL;DR

This paper proves a conjecture on the online Ramsey number for stars versus paths, establishing an exact formula that confirms the conjecture in a stronger form and advances understanding of online Ramsey theory.

Contribution

The paper proves the exact value of the online Ramsey number  r(K_{1,3}, P_{\u00b5}) for all b5 b5 2, confirming a conjecture and strengthening previous results.

Findings

 r(K_{1,3}, P_{b5}) = b5 d7 3/2 for all b5 b5 2

The conjecture by Latip and Tan is verified in a stronger form

Provides exact values for the online Ramsey number in this setting

Abstract

Given two graphs $G$ and $H$, the online Ramsey number $\tilde{r}(G,H)$ is defined to be the minimum number of rounds that Builder can always guarantee a win in the following $(G, H)$-online Ramsey game between Builder and Painter. Starting from an infinite set of isolated vertices, in each round Builder draws an edge between some two vertices, and Painter immediately colors it red or blue. Builder's goal is to force either a red copy of $G$ or a blue copy of $H$ in as few rounds as possible, while Painter's goal is to delay it for as many rounds as possible. Let $K_{1,3}$ denote a star with three edges and $P_{\ell}$ a path with $\ell$ vertices. Latip and Tan conjectured that $\tilde{r}(K_{1,3}, P_{\ell})=(3/2+o(1))\ell$ [Bull. Malays. Math. Sci. Soc. 44 (2021) 3511--3521]. We show that $\tilde{r}(K_{1,3}, P_{\ell})=\lfloor 3\ell/2 \rfloor$ for $\ell\ge 2$, which verifies the conjecture in a stronger form.