Let $k \in \mathbb{N}$ and let $H_1, H_2, \ldots, H_k$ be simple graphs such that for each $j \in \{ 1, 2, \ldots, k \}$, the vertex set of $H_j$ is $\{ 0, 1, 2, \ldots, n_j - 1 \}$ for some $n_j \in \mathbb{N}$. The ordered Ramsey number $R_\mathrm{ord}(H_1, H_2, \ldots, H_k)$ is the smallest $n \in \mathbb{N}$ for which every $k$-edge-coloring of the complete graph on the vertex set $\{ 0, 1, 2, \ldots, n - 1 \}$ contains $H_j$ as a monochromatic subgraph of color $j$ for some $j \in \{ 1, 2, \ldots, k \}$, with the vertices appearing in the same order as in $H_j$. Inspired by the work of Poljak, we apply the Kissat SAT solver to determine new small two-color ordered Ramsey numbers of various classes of graphs: monotone paths, monotone cycles, alternating paths, stars, complete graphs and nested matchings. In addition, we introduce the cyclic Ramsey numbers $R_\mathrm{cyc}(H_1, H_2, \ldots, H_k)$ as a natural relaxation of the ordered Ramsey numbers, and once again use Kissat to determine various such numbers for the two-color case. By observing structural patterns in the computational results, we determine all ordered or cyclic Ramsey numbers for several pairs of classes of graphs. Furthermore, we obtain some bounds on ordered and cyclic Ramsey numbers where one argument is a connected graph, while the other is a monotone path or a monotone cycle. We also explore how reinforcement learning can be used through the recently developed Reinforcement Learning for Graph Theory (RLGT) framework to obtain lower bounds on ordered and cyclic Ramsey numbers. Finally, we introduce the permutational Ramsey numbers to show how the different Ramsey-type formulations involving standard, ordered and cyclic Ramsey numbers can be unified within a group-theoretic framework.

翻译：设$k \in \mathbb{N}$，且$H_1, H_2, \ldots, H_k$为简单图，其中对每个$j \in \{ 1, 2, \ldots, k \}$，$H_j$的顶点集为$\{ 0, 1, 2, \ldots, n_j - 1 \}$（$n_j \in \mathbb{N}$）。有序拉姆齐数$R_\mathrm{ord}(H_1, H_2, \ldots, H_k)$是满足如下性质的最小整数$n \in \mathbb{N}$：对顶点集$\{ 0, 1, 2, \ldots, n - 1 \}$上的完全图进行$k$边染色时，必存在某个$j \in \{ 1, 2, \ldots, k \}$，使得染色图中包含一个颜色为$j$的$H_j$单色子图，且其顶点顺序与$H_j$中的顺序一致。受Poljak工作的启发，我们应用Kissat SAT求解器确定了多类图（单调路径、单调圈、交错路径、星图、完全图与嵌套匹配）的新的双色有序拉姆齐数。此外，我们引入循环拉姆齐数$R_\mathrm{cyc}(H_1, H_2, \ldots, H_k)$作为有序拉姆齐数的自然松弛，并再次利用Kissat求解器确定了双色情形下的若干此类数值。通过观察计算结果中的结构模式，我们确定了若干图类对的所有有序或循环拉姆齐数。进一步，我们获得了当一类参数为连通图而另一类为单调路径或单调圈时的有序与循环拉姆齐数的若干界。我们还探讨了如何通过近期发展的图论强化学习（RLGT, Reinforcement Learning for Graph Theory）框架，利用强化学习获得有序与循环拉姆齐数的下界。最后，我们引入置换拉姆齐数，以展示如何将涉及标准、有序与循环拉姆齐数的不同拉姆齐型表述统一在群论框架下。