Ramsey multiplicity for ordered graphs

Abstract

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 \(\blambda=(λ_1,\ldots,λ_k)\), 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 $\oM_{\boldsymbolλ} (n;\cS_{r_1,s_1},\cS_{r_2,s_2}) \le \min\{λ_1 B_{h_1}(n),λ_2 B_{h_2}(n)\}$ 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.