Free-cut decomposition for ordered matchings versus triangles

Definitions and mathematical statement

An ordered graph is a finite simple graph together with a linear order on its vertices. An ordered copy of $H$ in $G$ is an injective map $f:V(H)\to V(G)$ that preserves the vertex order and takes every edge of $H$ to an edge of $G$. Nonedges need not be preserved. For ordered graphs $H_1,H_2$, let $r_<(H_1,H_2)$ be the least $N$ such that every red--blue coloring of the complete ordered graph on $1<2<\cdots<N$ contains a red ordered copy of $H_1$ or a blue ordered copy of $H_2$.

Let $M$ range over all ordered matchings on $n$ vertices, with $n$ even: its edges are disjoint pairs that cover its vertices, but the endpoints can interleave arbitrarily in the prescribed order. The question is whether there are absolute constants $\varepsilon>0$ and $C<\infty$ such that \[ r_<(K_3,M)\le Cn^{2-\varepsilon} \] for every such $n$ and every ordering of every matching $M$. The order on the triangle has only one isomorphism type. Both constants must be independent of the particular matching and its ordering. A bound with an exponent depending on $M$ would not establish the assertion. The target is a uniform polynomial saving over a quadratic bound, even for highly crossing or nested arrangements of the matching edges.

Short English statement

Is there a uniform subquadratic bound for ordered Ramsey numbers of a triangle versus every ordered matching?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000953 (AIM-COMBINATORICS-0078), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Graph Ramsey theory, item 31. Original workshop question underlying AIM-COMBINATORICS-0078. https://www.overleaf.com/read/mnvcscjjysvg.