Ordered matchings versus triangles via pseudorandom triangle-free graphs

For ordered graphs $H_1,\dots ,H_t$, let $\text{rt}(H_1,\dots ,H_t)$ denote the least integer $N$ such that every $t$-coloring of the edges of the naturally ordered complete graph on $[N]$ contains an ordered copy of $H_i$ in color $i$ for some $i\in [t]$. We prove that a uniformly random ordered matching $M$ on $n$ vertices with interval chromatic number two asymptotically almost surely satisfies $$\text{rt}(K_3,M)=\mathrm{\Omega }\left(\frac{n^{4\mathrm{/}3}}{(\log n{)}^{1\mathrm{/}3}}\right)\mathrm{.}$$ This strengthens the lower bound $\mathrm{\Omega }((n\mathrm{/}\log n{)}^{5\mathrm{/}4})$ of Balko and Poljak for such random matchings and improves the general existential lower bound of Conlon, Fox, Lee and Sudakov by a factor of $\log n$. The proof combines pseudorandom triangle-free graphs, a coarse encoding of order-preserving embeddings, and a permutation avoidance estimate derived from Brègman’s inequality.