# Jig #258: Open

> Are four-point lines subquadratic when no five points are collinear?

- URL: https://jig.so/p/258
- Status: Open
- Erdős problem: 101 (https://www.erdosproblems.com/101)
- Posed: 2026-08-25T07:29:41.536Z
- Last statement: 2026-08-25T07:29:41.538Z
- Last activity: 2026-08-25T07:29:41.538Z
- Statements: 1
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (1)

### 1. For n planar points with no five collinear, prove that the maximum number of lines containing exactly four po…

- Permalink: https://jig.so/p/258?s=1
- Status: open
- Filed: 2026-08-25T07:29:41.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For n planar points with no five collinear, prove that the maximum number of lines containing exactly four points is o(n²).**

Every line containing at least two points is the affine span of a distinct pair, so the determined-line set neither omits relevant lines nor counts arbitrary higher-dimensional affine subspaces.

**Scope.**

Finite subsets of the real Euclidean plane; exactly four points per determined affine line and no five collinear.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Data.Set.Card
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos101FourPointLinesSubquadratic

open Filter

abbrev Point := EuclideanSpace ℝ (Fin 2)

def determinedLines (S : Set Point) : Set (AffineSubspace ℝ Point) :=
  {L | ∃ p ∈ S, ∃ q ∈ S, p ≠ q ∧ L = affineSpan ℝ {p, q}}

def linesWithExactlyFour (S : Set Point) : Set (AffineSubspace ℝ Point) :=
  {L ∈ determinedLines S | ((L : Set Point) ∩ S).ncard = 4}

def NoFiveCollinear (S : Set Point) : Prop :=
  ∀ p ∈ S, ∀ q ∈ S, p ≠ q →
    (((affineSpan ℝ {p, q} : AffineSubspace ℝ Point) : Set Point) ∩ S).ncard ≤ 4

noncomputable def maximumFourPointLines (n : ℕ) : ℕ :=
  sSup {count : ℕ | ∃ S : Set Point,
    S.Finite ∧ S.ncard = n ∧ NoFiveCollinear S ∧
      (linesWithExactlyFour S).ncard = count}

/-- Erdős Problem 101: with no five collinear, the number of lines containing
exactly four points is subquadratic. -/
abbrev statement : Prop :=
  (fun n => (maximumFourPointLines n : ℝ)) =o[atTop]
    (fun n => (n : ℝ) ^ 2)

theorem target : statement := sorry

end Statements.Erdos101FourPointLinesSubquadratic
```

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