A Note on Restricted Online Ramsey Numbers of Matchings

TL;DR

This paper determines the exact restricted online Ramsey numbers for matchings in specific cases, confirming previous bounds and providing new lower bounds for these numbers.

Contribution

It establishes the precise values of restricted online Ramsey numbers for matchings when n=3r-1 and n'=4r-2, answering an open question from prior research.

Findings

Confirmed that r-1 is the exact value for r-1 cases.

Provided a new lower bound for n'=4r-2 cases.

Validated previous upper bounds as tight for certain parameters.

Abstract

The restricted online Ramsey numbers were introduced by Conlon, Fox, Grinshpun and He in 2019. In a recent paper, Briggs and Cox studied the restricted online Ramsey numbers of matchings and determined a general upper bound for them. They proved that for $n=3r-1=R_2(r K_2)$ we have $\tilde{R}_{2}(r K_2;n) \leq n-1$ and asked whether this was tight. In this short note, we provide a general lower bound for these Ramsey numbers. As a corollary, we answer this question of Briggs and Cox, and confirm that for $n=3r-1$ we have $\tilde{R}_{2}(r K_2;n) = n-1$. We also show that for $n'=4r-2=R_3(r K_2)$ we have $\tilde{R}_{3}(r K_2;n') = 5r-4$.