Suppose we begin with a set of points S in the plane

Definitions and mathematical statement

For a finite set $S\subset\mathbb R^2$, let $\mathcal L(S)$ be the set of distinct lines determined by pairs of distinct points of $S$. In the Euclidean interpretation define \[ T(S)=\{x\in\mathbb R^2:\text{there exist distinct nonparallel } \ell,\ell'\in\mathcal L(S)\text{ with }x=\ell\cap\ell'\}. \] Coincident lines are one member of $\mathcal L(S)$, and parallel lines add no point at infinity. Starting with $S_0=S$, set $S_{i+1}=T(S_i)$ and $N_i=|S_i|$ for integers $i\geq0$.

Problem. Determine how rapidly $N_i$ grows under repeated line-and-intersection construction, including the dependence on the initial configuration $S$. The source does not specify the size or nondegeneracy of $S$, whether it seeks universal bounds, a typical-configuration rate, or a classification of possible rates. Finite Euclidean sets are the explicit conventional setting above; an infinite or projective-plane version would be a different question.

The operator is the source's replacement $T(S)$, not automatically the accumulated closure $S\cup T(S)$. For degenerate initial sets those definitions can differ, so initial-point retention must not be added silently. No general-position assumption is imposed.

The supplied research summary describes doubly exponential growth as a candidate maximal scale and says the general exact growth question remains unresolved. It mentions rank/lattice-degree arguments, Sylvester--Gallai configurations, and degeneracies only as context, without a precise bound or theorem to incorporate.

Short English statement

Study the possible growth rates obtained by repeatedly intersecting all lines determined by the current finite planar point set. The source leaves the initial-position assumptions and the requested type of growth bound unspecified.

Sources

• Cooper, Combinatorial Problems I Like, Geometry, source-order bullet 5, October 2020 snapshot, https://people.math.sc.edu/cooper/combprob.html; HTML extraction, accessed 2026-07-29. The author page labels the item open in that snapshot; later resolutions were not checked item by item according to the record.
• The statement has a see-this link whose target is absent from the extraction. The research summary names Magnuson and collaborators and uses the phrase Hetzheim-type growth, but supplies no identifiable publication, theorem, or discussion URL.
• Local registry data/unsolvedmath/problems.json, permanent ID $3100005$, source code catalogue entry, Suppose we begin with a set of points S in the plane. The record labels this open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications.