Ramsey multiplicity for ordered graphs

Let $\cG_1,\ldots,\cG_k$ be fixed vertex-ordered graphs, each containing at least one edge. The ordered Ramsey number $\oR(\cG_1,\ldots,\cG_k)$ is the least integer $N$ such that every $k$-edge-coloring of the ordered complete graph $\cK_N$ contains an order-preserving copy of $\cG_i$ in color $i$ for some $i\in[k]$. For positive weights $\blambda=(\lambda_1,\ldots,\lambda_k)$, let $\oM_{\blambda}(n;\cG_1,\ldots,\cG_k)$ denote the minimum weighted number of correctly colored, order-preserving copies of the target graphs over all $k$-edge-colorings of $\cK_n$. When $\blambda=\bf{1}$, $\oM_{\bf{1}}(n;\cG_1,\ldots,\cG_k)=\oM(n;\cG_1,\ldots,\cG_k)$ is called the ordered Ramsey multiplicity. In this paper, we first establish the amplification inequality $$\oM_{\blambda}(n;\cG_1,\ldots,\cG_k) \ge \oM_{\blambda}(t;\cG_1,\ldots,\cG_k) \frac{\binom{n}{\hmin}}{\binom{t}{\hmin}},$$ where $h_i=v(\cG_i),\hmin=\min_{i\in[k]}h_i$, and $n\ge t\ge\oR(\cG_1,\ldots,\cG_k)$. Let $\cS_{r,s}$ be the ordered star whose center has $r-1$ leaves to its left and $s-1$ leaves to its right, and let $\bB_m$ be the family of all ordered perfect matchings on $[2m]$ containing the edge $\{1,2m\}$. We apply the amplification inequality to obtain the multiplicity lower bounds for ordered stars and ordered perfect matchings. We then obtain the upper bound $\oM_{\boldsymbol\lambda} (n;\cS_{r_1,s_1},\cS_{r_2,s_2}) \le \min\{\lambda_1 B_{h_1}(n),\lambda_2 B_{h_2}(n)\}$ by constructions, where $B_{h_i}(n):= \binom{\lfloor n/2\rfloor}{h_i} + \binom{\lceil n/2\rceil}{h_i}$ and $h_i=r_i+s_i-1$ for $i\in [2]$. We also derive a random-coloring upper bound for ordered stars and prove $$\oM(n; \bB_m,\bB_m) \le \binom{n}{2m} \frac{(2m-2)!}{2^{2m-2}(m-1)!}.$$ Finally, we establish a regularity-based lifting theorem for ordered colorings.