Lower and Upper Bounds for Small Canonical and Ordered Ramsey Numbers

Abstract: In this paper, we investigate three extensions of Ramsey numbers to other combinatorial settings. We first consider ordered Ramsey numbers. Here, we ask for a monochromatic copy of a linearly ordered graph $G$ in every $2$-edge-coloring of a linearly ordered complete graph $K_n$. The smallest such $n$ is denoted by $\vec{R}(G)$. Next, we study canonical Ramsey numbers. A canonical coloring of a linearly ordered graph $G$ is an edge-coloring in which $G$ is monochromatic, rainbow, or min/max-lexicographic. In the latter case, each pair of edges receives the same color if and only if they share the same first (respectively, second) vertex. Erd\H{o}s and Rado showed that for every $p$ there exists $n$ such that every edge-coloring of a linearly ordered $K_n$ contains a canonical copy of $K_p$; the smallest such $n$ is denoted by $ER(G)$. Finally, we examine unordered canonical Ramsey numbers, introduced by Richer. An edge-coloring of $G$ is orderable if there exists a linear ordering of its vertices such that the color of each edge is determined by its first vertex. Unlike lexicographic colorings, this notion also includes monochromatic colorings. Richer proved that for all $s$ and $t$, there exists $n$ such that every edge-coloring of $K_n$ contains an orderable copy of $K_s$ or a rainbow $K_t$. The smallest such $n$ is denoted by $CR(s,t)$. In all three settings, we focus on determining the corresponding Ramsey numbers for small graphs $G$. We use tabu search and integer programming to obtain lower bounds, and flag algebras or integer programming to establish upper bounds. Among other results, we determine $\vec{R}(G)$ for all graphs $G$ on up to four vertices except $K_4^-$, $ER(P_4)$ for all orderings of $P_4$, and the exact values $CR(6,3)=26$ and $CR(3,5)=13$.