A linear worst-case lower bound for Ramsey numbers of rectilinear knots

Definitions and mathematical statement

A tame knot is an embedded circle in $\mathbb R^3$ considered up to ambient isotopy. Its crossing number $\operatorname{cr}(K)$ is the smallest number of crossings in any regular planar knot diagram. A rectilinear embedding of the complete graph $K_N$ places $N$ distinct vertices in $\mathbb R^3$ and realizes each edge by its straight segment, with distinct edges meeting only at their common endpoints. Each graph cycle is consequently a polygonal knot.

Let $R_{\rm knot}(K)$ be the least $N$ such that every such embedding contains a cycle ambient isotopic to $K$ or to its mirror image. This is the source's ambient-homeomorphism convention for containment, allowing a reflection of three-dimensional space. Define \[ R_{\rm knot}(k)=\max_{\operatorname{cr}(K)\leq k}R_{\rm knot}(K). \] The maximum is over the finitely many knot types with at most $k$ crossings, and is well-defined using the knot-containment theorem quoted in the workshop. Determine the growth of $R_{\rm knot}(k)$ as $k\to\infty$, with upper bounds valid for every geometric placement and lower bounds furnished by placements avoiding a prescribed knot type. Containment requires a single cycle in the embedded graph, not a drawing obtained after moving its vertices. The oriented ambient-isotopy variant can also be considered by forbidding the mirror alternative; its convention must be stated separately. The central issue is quantitative dependence on knot complexity, rather than the qualitative existence of a finite threshold.

Short English statement

How large must a straight-line complete graph in space be to force every knot of a prescribed crossing complexity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000936 (AIM-COMBINATORICS-0061), 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 14. Original workshop question underlying AIM-COMBINATORICS-0061. https://www.overleaf.com/read/mnvcscjjysvg.