Ordered matchings versus triangles via pseudorandom triangle-free graphs

Abstract: For ordered graphs $H_1,\ldots,H_t$, let $\rt(H_1,\ldots,H_t)$ 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 [ \rt(K_3,M) =Ω\left(\frac{n^{4/3}}{(\log n)^{1/3}}\right). ] This strengthens the lower bound $Ω((n/\log n)<sup>{5/4})$ 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.