Big-line-big-clique conjecture

Determine whether, for every pair of integers k,ℓ≥3, every sufficiently large finite planar point set contains either ℓ collinear points or k pairwise visible points.

Background

The paper studies visibility graphs of finite planar point sets, where two points are adjacent when the open segment joining them contains no other point of the set. The big-line-big-clique conjecture predicts a dichotomy between large collinear subsets and large cliques in the visibility graph.

The paper records that the conjecture is known for several ranges of parameters, including k≤5 and the recently resolved case (k,ℓ)=(6,4), but that the next case (7,4) remains unresolved. The authors’ Corollary 1 comes within one visibility edge of this case by guaranteeing seven points with at most one non-visible pair.

References

The celebrated big-line-big-clique conjecture of \citet{KaraPorWood} asserts that, for every pair of integers $k,\ell\ge3$, every sufficiently large finite planar point set contains either $\ell$ collinear points or $k$ pairwise visible points (see and references therein for other related results and conjectures).

— Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results  (2609.25727 - Bhattacharya et al., 22 Sep 2026) in Section 1, Introduction

This result comes within a single visibility edge of the next open case, $(k,\ell)=(7,4)$, of the big-line-big-clique conjecture.

— Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results  (2609.25727 - Bhattacharya et al., 22 Sep 2026) in Remark following Corollary 1, Section 1

Determining the exact minimum number of points required remains open.

— Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results  (2609.25727 - Bhattacharya et al., 22 Sep 2026) in Remark following Corollary 4, Appendix A