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OP-8834OpenGeometry and topology

Erdős #101 · geometry

Given nn points in R2\mathbb{R}^2, no five of which are on a line, the number of lines containing four points is o(n2)o(n^2).

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Given nn points in R2\mathbb{R}^2, no five of which are on a line, the number of lines containing four points is o(n2)o(n^2).

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Background

There are examples of sets of nn points with ∼n2/6\sim n^2/6 many collinear triples and no four points on a line. Such constructions are given by Burr, Grünbaum, and Sloane [BGS74] and Füredi and Palásti [FuPa84].

Grünbaum [Gr76] constructed an example with ≫n3/2\gg n^{3/2} such lines. Erdős speculated this may be the correct order of magnitude. This is false: Solymosi and Stojaković [SoSt13] have constructed a set with no five on a line and at least

n2−O(1/log⁡n)n^{2-O(1/\sqrt{\log n})}

many lines containing exactly four points.

See also [102] and [669]. A generalisation of this problem is asked in [588].

This problem is Problem 71 on Green's open problems list.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #101 link
  2. [2]Upstream Lean 4 statement for Erdős #101 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 101 link

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