Off-diagonal convex geometric Ramsey bounds

Definitions and mathematical statement

The entire original problem is "Improve these bounds," under "Ordered Ramsey numbers." The antecedent bounds and parameter definitions are absent. The attached aid identifies the intended target as off-diagonal convex geometric Ramsey numbers; this is a supplied reconstruction, not established by the extracted sentence alone.

For a precise interpretation, take planar point sets in general position (no three collinear). Let $\operatorname{ES}(k)$ be the least size forcing $k$ points in convex position. Let $g(s,n)$ be the least $N$ such that every red/blue edge-coloring of the complete geometric graph on any such $N$ points contains either a red complete graph on $s$ convex-position points or a blue one on $n$ convex-position points. Let $r(a,b)$ be the ordinary red/blue Ramsey number. For $s,n\ge3$ put \[ T=(s-1)^2+(n-1)^2+1. \] The aid reports \[ (\operatorname{ES}(s)-1)(\operatorname{ES}(n)-1)+1\le g(s,n) \le\min\left\{r(\operatorname{ES}(s),\operatorname{ES}(n)),\ (T+1)2^T T!\right\}. \] It attributes the lower construction to an order-type-preserving geometric cluster product and the explicit upper refinement to optimized asymmetric thresholds in a Mubayi--Suk canonical-selection argument.

The stated consequences are \[ \log_2g(n,n)\le(4+o(1))n^2\log_2n\quad(n\to\infty), \] and, for each fixed $s\ge3$, \[ g(s,n)\le\frac{2^{(\operatorname{ES}(s)-1)n+O_s(\sqrt{n\log n})}}{n^{\operatorname{ES}(s)-2}}. \] In particular it gives \[ 2^{n-1}+1\le g(3,n)\le\frac{2^{2n+O(\sqrt{n\log n})}}n. \] The main diagonal exponential gap is recorded as still open. The exact earlier bounds the original requested improving, and the precise canonical-selection theorem being sharpened, are not supplied here. The formulas therefore describe the aid's partial result rather than a fully source-verified resolution of the fragment.

Short English statement

The original fragment lacks the bounds it asks to improve. The supplied aid reconstructs a convex geometric Ramsey problem and gives explicit product lower and factorial upper bounds, including diagonal and fixed-parameter consequences, while leaving the main gap open.

Sources

• Local UnsolvedMath registry, permanent ID $20000955$, source code catalogue entry; canonical aim-combinatorics-notes.json, notes[79].
• AIM "Graph Ramsey theory," section "Ordered Ramsey numbers," source item 33: https://www.overleaf.com/read/mnvcscjjysvg. The exact extracted problem consists only of "Improve these bounds." No original remarks or bibliography are supplied.
• Attached research attempt 1: partial result, run and registry status partially solved, rigor proved, novelty confidence low, classification PARTIAL-PROGRESS. It names Erdős--Szekeres numbers and Mubayi--Suk without exact references. This machine-generated, non-peer-reviewed aid requires independent verification before citation, including its reconstructed target.