论文标题
关于限制在线拉姆齐的注释
A Note on Restricted Online Ramsey Numbers of Matchings
论文作者
论文摘要
受限的在线拉姆齐号码是由Conlon，Fox，Grinshpun和He引入的。在最近的一篇论文中，Briggs和Cox研究了受限制的在线Ramsey比赛数量，并确定了他们的一般上限。他们证明，对于$ n = 3r-1 = r_2（r k_2）$，我们有$ \ tilde {r} _ {2} _ {2}（r k_2; n）\ leq n-1 $，并问这是否很紧。在此简短说明中，我们为这些拉姆齐号提供了一般的下限。作为推论，我们回答了Briggs和Cox的这个问题，并确认对于$ n = 3r-1 $，我们有$ \ tilde {r} _ {2}（r k_2; n）= n-1 $。我们还表明，对于$ n'= 4R-2 = r_3（r k_2）$，我们有$ \ tilde {r} _ {3}（r k_2; n'）= 5r-4 $。
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 $\tilde{R}_{3}(r K_2;n') = 5r-4$.
