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Updated Oct 7

Problem 102

Statement. Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\mathbb{R}^2$ such that there are $\geq cn^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$ . Is it true that, for fixed $c>0$ , we have $h_c(n)\to \infty$ ?

Status. Open.

Source. erdosproblems.com/102, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #102, https://www.erdosproblems.com/102.

References.

[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (3)

Discrete and Convex Geometry discrete_geometry/currier_2025_more_pointsets_many_rich_lines Theorem 1.3 (p. 2): a product of two nice-basis progressions has Omega(n^2/r^3) r-rich lines for every r <= C' n^alpha

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