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arXiv 0902.1715math.CO
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在线Ramsey数

On-line Ramsey numbers

David Conlon

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11 pages
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摘要：

考虑以下两个玩家之间的游戏，建筑师和画家。Builder每次绘制一个边，Painter根据每个边的出现将其着色为红色或蓝色。Builder的目的是迫使Painter绘制固定图G的单色副本。不管Painter的策略如何，Builder必须绘制的最小边数被称为G的在线Ramsey数\tilde{r}（G）。关于\tilde{r}（K_t）= o（\binom{r(t)}{2}）的猜想，我们的主要结果是：存在一个常数c > 1，使得对于无穷多个t值，\tilde{r}（K_t）\leq c^{-t} \binom{r(t)}{2}.我们还证明了这个数的一个更具体的上界，证明了存在一个常数c使得\tilde{r}（K_t）\leq t^{-c \frac{\log t}{\log \log t} } 4 ^t。最后，我们证明了完全二部图K_{t,t}的在线Ramsey数的一个新上界。

英文摘要：

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}.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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