Point Incidence on One-Lipschitz Graphs

Definitions and mathematical statement

The cleaned source asks: given a finite set $P$ of points in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$, what is the maximum number of these points lying on the graph of a $1$-Lipschitz function?

A precise conventional model, conditional on using the last coordinate as height and Euclidean norms, is the following. For $d\in\{2,3\}$, write ${\mathbb{R}}^d={\mathbb{R}}^{d-1}\times{\mathbb{R}}$. A function $f:{\mathbb{R}}^{d-1}\to{\mathbb{R}}$ is $1$-Lipschitz if \[ |f(x)-f(y)|\leq\|x-y\|_2\quad\hbox{for all }x,y\in{\mathbb{R}}^{d-1}. \] Determine, from the given finite set $P\subset{\mathbb{R}}^d$, the quantity \[ M(P)=\max_{\operatorname{Lip}(f)\leq1} \left|P\cap\{(x,f(x)):x\in{\mathbb{R}}^{d-1}\}\right|. \] The extraction does not specify the height direction, norm, domain of $f$, or whether an algorithm, deterministic bound, or random-model asymptotic is intended. The displayed model therefore explains one reading rather than fixing these missing choices as source hypotheses. The original also says "finite distribution or set"; a weighted or random distribution would need its own objective and probability law. No such distribution is supplied.

Short English statement

For a given finite planar or spatial point set, maximize the number of points on a $1$-Lipschitz graph. The natural finite-set optimization can be written down once the graph direction and metric are fixed, but the source's distributional and boarding-policy context is incomplete.

Sources

• Local UnsolvedMath registry, permanent ID $30000995$, source code catalogue entry, "Point Incidence on One-Lipschitz Graphs." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Calculus of Variations (2009), Oberwolfach Reports, pp. 1707--1770, DOI https://doi.org/10.4171/owr/2008/31.
• Supplied literature triage, literature_checked_at: 2026-08-21: labeled open, but says the extracted passage stops before specifying the boarding-policy distribution or optimization problem and is incomplete rather than a verifiable solved claim. This is dated registry curation and requires independent review; its boarding-policy wording does not match the complete finite-set optimization suggested by the cleaned statement.