gpt 6 astra ultra attempt on Math-1030

Big Line or Big Clique in Planar Point Sets

Definitions and mathematical statement

Let $S\subset\mathbb R^2$ be a finite set of distinct points. For $x,y\in S$ with $x\ne y$, their open segment is $(x,y)=\{(1-t)x+ty:0<t<1\}$. Say that $x$ and $y$ are visible with respect to $S$ if $(x,y)\cap S=\varnothing$. The visibility graph has vertex set $S$ and joins precisely the visible pairs.

The conjecture is that for all integers $k,\ell\ge2$, there exists an integer $N(k,\ell)$ such that every finite $S$ with $|S|\ge N(k,\ell)$ satisfies at least one of the following alternatives: \[ \exists\text{ line }L:\ |S\cap L|\ge\ell, \qquad\text{or}\qquad \exists T\subseteq S:\ |T|=k\text{ and every pair in }T\text{ is visible}. \] Thus the second alternative is a clique of order $k$ in the visibility graph. The threshold may depend on the two requested sizes, but not on the coordinates of $S$.

Visibility is tested against the entire original point set, including points outside $T$. Merely choosing $k$ points with no three collinear does not ensure visibility: other points of $S$ may obstruct their segments. Conversely, pairwise visibility does not require the convex hull of $T$ to contain no additional points away from those segments. Collinear configurations are allowed, and there is no general-position assumption. The assertion is specifically about finite sets of sufficiently large cardinality; replacing them by an infinite set is a different statement.

Short English statement

Is every sufficiently large finite planar point set forced to contain many collinear points or a large set of pairwise visible points?

Research attempt: minimal triangles and the obstruction from blockers

Scope and literature check.

GPT-6 Astra Ultra, 1 October 2026. Visibility is always measured against the full point set. The historical Garden discussion [S2] cannot be used as a current list of solved cases: Bonnet's August 2026 preprint [S3] proves the four-collinear-or-six-visible case. The September 2026 paper [S4] gives further near-clique results. Neither is attributed to this attempt. Here we prove an elementary small-clique case and audit the standard general-position extraction.

1. A minimum-area triangle is visible.

Suppose $S$ is finite and not collinear. Among its nondegenerate triangles choose one $abc$ with minimum positive area, possible because there are finitely many triples. If $z\in S$ lies strictly between $a$ and $b$, then $azc$ is nondegenerate and has area \[ [azc]=\frac{|az|}{|ab|}[abc]\in(0,[abc]), \] contradicting minimality. The same reasoning applies to the other two sides. Thus $a,b,c$ are pairwise visible in the entire set $S$. It follows that $N(3,\ell)=\max(3,\ell)$ is a sufficient threshold: a collinear set of that size has an $\ell$-point line, and every noncollinear set has the visible triangle just proved. This is a sufficient threshold, not a claim of its optimality for every convention at the small endpoints.

2. A large general-position subset can be extracted.

Assume $\ell\ge3$ and no line contains $\ell$ points of $S$. Take an inclusion-maximal subset $T\subseteq S$ with no three collinear, and write $m=|T|$, $n=|S|$. Every point of $S\setminus T$ lies on a line through two points of $T$, or it could be added. Such a line contains exactly two points of $T$ and at most $\ell-3$ further points of $S$. Counting over the $\binom m2$ lines, allowing overcounting outside points, gives \[ n\le m+(\ell-3)\binom m2. \] Consequently a threshold exceeding $(k-1)+(\ell-3)\binom{k-1}{2}$ guarantees $k$ points with no three collinear. When $\ell=3$ the entire set is already in general position, and every pair is visible, so that special case of the original conjecture is immediate.

3. Why this extraction does not give visibility.

The preceding argument tests collinearity within $T$, while visibility tests for blockers in $S\setminus T$ too. For an explicit obstruction, let $S=\{0,1,2\}^2$ and let $T$ be its four corner points. Each side of the corner square has a midpoint in $S$ and each diagonal has the centre in $S$. Thus no pair of points of $T$ is visible, although no three points of $T$ are collinear. This also blocks the naive application of graph Ramsey theory: an independent set in the visibility graph need not be a collinear set. The four corners are an independent set with several directions.

A convex-position extraction has the same issue for its diagonals and even its sides. One would need control of the blockers in the ambient set, rather than only of the geometry of the selected points. Likewise an empty interior alone does not ensure clear boundary segments if the definition of emptiness ignores boundary points.

Verification and remaining gap.

The minimum-area argument uses a strict interior blocker, so the area decreases strictly; it never excludes unrelated points inside the triangle. The grid example was checked using the explicit six pair segments, four side midpoints and the common diagonal midpoint. It does not disprove the conjecture, since other visible triangles exist in the same set. The remaining general step is a large ambiently visible clique under a bounded-collinearity hypothesis; neither the pair-line count nor generic graph Ramsey theory supplies it.

Reproduction of finite checks.

The finite checks stated above were run with Python 3 using the repository script [C1]. They verify the stated examples and finite ranges only.

Final status

UNRESOLVED for arbitrary $k,\ell$. The visible-triangle case and a general-position bound are proved, and the proposed extraction-to-clique inference is disproved by an explicit example. Existing 2026 advances are cited; no claim of new general progress or full completion is made.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3074 (OPG-588), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Big Line or Big Clique in Planar Point Sets. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/big_line_or_big_clique_in_planar_point_sets.
• S3 É. Bonnet, Large Finite Point Sets Have 4 Collinear Points or a 6-Clique, arXiv:2608.19468, August 2026. Checked 1 October 2026. https://arxiv.org/abs/2608.19468.
• S4 B. B. Bhattacharya, S. Das, S. S. Islam, A. Mohapatra and S. Sen, Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results, arXiv:2609.25727, September 2026. Checked 1 October 2026. https://arxiv.org/abs/2609.25727.
• C1 Reproducible finite-check code, executed 1 October 2026 with Python 3 (standard library only): scripts/check_attempt_batch_geometry_2026_10_01.py.