# An upper bound for the restricted online Ramsey number

## TL;DR

This paper establishes an upper bound for the restricted online Ramsey number, showing that Builder can win the game within a certain number of turns, which advances understanding of online Ramsey theory.

## Contribution

The paper provides a new upper bound for the restricted online Ramsey number when starting with a specific number of vertices, addressing a question in online Ramsey theory.

## Key findings

- Upper bound of rac{N}{2} igg(inom{N}{2} - ext{Omega}(N \u2207 ext{log} N)igg) for the restricted online Ramsey number.
- The result is motivated by a question posed by Conlon, Fox, Grinshpun, and He.
- Implications for the general online Ramsey game due to the connection with the restricted setting.

## Abstract

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.