  The restricted $(m,n;N)$-online Ramsey game is a game played between two
players, Builder and Painter. The game starts with $N$ isolated vertices. Each
turn Builder picks an edge to build and Painter chooses whether that edge is
red or blue, and Builder aims to create a red $K_m$ or blue $K_n$ in as few
turns as possible. The restricted online Ramsey number $\tilde{r}(m,n;N)$ is
the minimum number of turns that Builder needs to guarantee her win in the
restricted $(m,n;N)$-online Ramsey game. We show that if $N=r(n,n)$, \[
\tilde{r}(n,n;N)\le \binom{N}{2} - \Omega(N\log N), \] motivated by a question
posed by Conlon, Fox, Grinshpun and He. The equivalent game played on
infinitely many vertices is called the online Ramsey game. As almost all known
Builder strategies in the online Ramsey game end up reducing to the restricted
setting, we expect further progress on the restricted online Ramsey game to
have applications in the general case.

Combinatorics