A Note on Restricted Online Ramsey Numbers of Matchings

Vojtěch Dvořák

paper

A Note on Restricted Online Ramsey Numbers of Matchings

arXiv:2010.04113

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 $n = 3 r - 1 = R_{2} (r K_{2})$ we have $R ~_{2} (r K_{2}; n) \leq n - 1$ 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 $n = 3 r - 1$ we have $R ~_{2} (r K_{2}; n) = n - 1$ . We also show that for $n^{'} = 4 r - 2 = R_{3} (r K_{2})$ we have $R ~_{3} (r K_{2}; n^{'}) = 5 r - 4$ .

3 pages