Ramsey multiplicity for ordered graphs

Let ${\text{cG}}_1,\dots ,{\text{cG}}_k$ be fixed vertex-ordered graphs, each containing at least one edge. The ordered Ramsey number $\text{oR}({\text{cG}}_1,\dots ,{\text{cG}}_k)$ is the least integer $N$ such that every $k$-edge-coloring of the ordered complete graph ${\text{cK}}_N$ contains an order-preserving copy of ${\text{cG}}_i$ in color $i$ for some $i\in [k]$. For positive weights $\text{blambda}=({\lambda }_1,\dots ,{\lambda }_k)$, let ${\text{oM}}_{\text{blambda}}(n;{\text{cG}}_1,\dots ,{\text{cG}}_k)$ denote the minimum weighted number of correctly colored, order-preserving copies of the target graphs over all $k$-edge-colorings of ${\text{cK}}_n$. When $\text{blambda}=\mathbf{1}$, ${\text{oM}}_{\mathbf{1}}(n;{\text{cG}}_1,\dots ,{\text{cG}}_k)=\text{oM}(n;{\text{cG}}_1,\dots ,{\text{cG}}_k)$ is called the ordered Ramsey multiplicity. In this paper, we first establish the amplification inequality $${\text{oM}}_{\text{blambda}}(n;{\text{cG}}_1,\dots ,{\text{cG}}_k)\ge {\text{oM}}_{\text{blambda}}(t;{\text{cG}}_1,\dots ,{\text{cG}}_k)\frac{\left(\genfrac{}{}{0px}{}{n}{\text{hmin}}\right)}{\left(\genfrac{}{}{0px}{}{t}{\text{hmin}}\right)},$$ where $h_i=v({\text{cG}}_i),\text{hmin}={\min }_{i\in [k]}{h}_i$, and $n\ge t\ge \text{oR}({\text{cG}}_1,\dots ,{\text{cG}}_k)$. Let ${\text{cS}}_{r,s}$ be the ordered star whose center has $r-1$ leaves to its left and $s-1$ leaves to its right, and let ${\text{bB}}_m$ 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 ${\text{oM}}_{\mathit{\lambda }}(n;{\text{cS}}_{r_1,s_1},{\text{cS}}_{r_2,s_2})\le \min \{{\lambda }_1{B}_{h_1}(n),{\lambda }_2{B}_{h_2}(n)\}$ by constructions, where $B_{h_i}(n):=\left(\genfrac{}{}{0px}{}{\lfloor n\mathrm{/}2\rfloor }{h_i}\right)+\left(\genfrac{}{}{0px}{}{\lceil n\mathrm{/}2\rceil }{h_i}\right)$ 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 $$\text{oM}(n;{\text{bB}}_m,{\text{bB}}_m)\le \left(\genfrac{}{}{0px}{}{n}{2m}\right)\frac{(2m-2)!}{2^{2m-2}(m-1)!}\mathrm{.}$$ Finally, we establish a regularity-based lifting theorem for ordered colorings.