AI 中文总结
该研究针对星图与长路、长圈的在线拉姆齐数问题，改进了固定k≥4时两类在线拉姆齐数的渐近上界，将原有的(k+o(1))n提升至更紧的((2k+4)/5+o(1))n，推动了在线拉姆齐理论的定量研究。
AI 中文摘要
图G和H的在线拉姆齐博弈在无穷完全图$K_\mathbb{N}$上进行。每一轮中，Builder（构建者）选择一条边，Painter（着色者）将其染为红色或蓝色。在线拉姆齐数$\tilde{r}(G,H)$是满足Builder有策略保证在t轮内得到红色G副本或蓝色H副本的最小整数t。对每个固定整数$k\ge4$，当$n\to\infty$时，$\tilde{r}(K_{1,k},P_n)$和$\tilde{r}(K_{1,k},C_n)$的已知最优下界为$\left(\frac{k+3}{4}+o(1)\right)n$。我们将对应的渐近上界从$(k+o(1))n$改进为当$n\to\infty$时的$\left(\frac{2k+4}{5}+o(1)\right)n$。
英文摘要
The online Ramsey game for graphs $G$ and $H$ is played on the infinite complete graph $K_\mathbb{N}$. In each round, Builder chooses an edge, and Painter colors it red or blue. The online Ramsey number $\tilde{r}(G,H)$ is the smallest integer $t$ for which Builder has a strategy guaranteeing a red copy of $G$ or a blue copy of $H$ within $t$ rounds. For every fixed integer $k\ge4$, the best-known lower bounds for $\tilde{r}(K_{1,k},P_n)$ and $\tilde{r}(K_{1,k},C_n)$ are $\left(\frac{k+3}{4}+o(1)\right)n$ as $n\to\infty$. We improve the corresponding asymptotic upper bounds from $(k+o(1))n$ to $\left(\frac{2k+4}{5}+o(1)\right)n$ as $n\to\infty$.
