On-line Ramsey numbers

TL;DR

This paper investigates the on-line Ramsey number for complete and bipartite graphs, establishing new upper bounds and supporting conjectures about its growth rate relative to classical Ramsey numbers.

Contribution

It provides novel upper bounds for the on-line Ramsey number of complete graphs and bipartite graphs, advancing understanding of its asymptotic behavior.

Findings

Existence of a constant c > 1 such that  r(K_t)  ^{-t} inom{r(t)}{2} for infinitely many t

Upper bound  r(K_t)   t^{-rac{ ext{log} t}{ ext{log} ext{log} t}} 4^t

New upper bound for on-line Ramsey number of K_{t,t}

Abstract

Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colours them, in either red or blue, as each appears. Builder's aim is to force Painter to draw a monochromatic copy of a fixed graph G. The minimum number of edges which Builder must draw, regardless of Painter's strategy, in order to guarantee that this happens is known as the on-line Ramsey number \tilde{r}(G) of G. Our main result, relating to the conjecture that \tilde{r}(K_t) = o(\binom{r(t)}{2}), is that there exists a constant c > 1 such that \tilde{r}(K_t) \leq c^{-t} \binom{r(t)}{2} for infinitely many values of t. We also prove a more specific upper bound for this number, showing that there exists a constant c such that \tilde{r}(K_t) \leq t^{-c \frac{\log t}{\log \log t}} 4^t. Finally, we prove a new upper bound for the on-line Ramsey number of the complete bipartite graph K_{t,t}.