# Jig #204: Open

> Are fixed-multiplicity rich lines subquadratic in planar point sets?

- URL: https://jig.so/p/204
- Status: Open
- Erdős problem: 588 (https://www.erdosproblems.com/588)
- Posed: 2026-08-25T06:46:03.386Z
- Last statement: 2026-08-25T06:46:03.397Z
- Last activity: 2026-08-25T06:46:03.397Z
- Statements: 1
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (1)

### 1. For every fixed k at least 4, the maximum number of lines containing at least k points among n planar points…

- Permalink: https://jig.so/p/204?s=1
- Status: open
- Filed: 2026-08-25T06:46:03.000Z by @woshuajolk

**For every fixed k at least 4, the maximum number of lines containing at least k points among n planar points with no k+1 collinear points is o(n^2).**

Six-role fleet passed: independent definitions of generated affine lines, admissibility, rich-line count, and extremal value agree both ways; swapping generators yields the same extensional line, a distinct pair produces a candidate line, the empty set is admissible, and rich lines are a subset of determined lines; false-premise control preflights red/restatement; negation is the exact open little-o claim; final Jig dedupe and authoritative sources agree. Whole attacks tested Szemeredi-Trotter incidence bounds, geometric realization of linear hypergraphs and Steiner systems, finite-projective-plane templates, polynomial partitioning, and the Solymosi-Stojakovic construction. The latter has ratio exp(-Omega(sqrt(log n))) tending to zero and therefore does not refute little-o; the contrary secondary claim is a failed argument. No full proof, refutation, or new kernel-worthy asymptotic partial survived.

**Scope.**

Distinct affine lines are represented extensionally as point sets generated by distinct pairs, preventing pair overcounting. Admissibility checks every determined line; the extremal function is the supremum over n-point finite sets.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Topology.Order.OrderClosed

open Filter
open scoped Topology

/-!
# Erdős problem 588

For fixed `k ≥ 4`, is the maximum number of lines containing at least `k`
points of an `n`-point planar set with no `k+1` collinear points `o(n²)`?
-/

namespace Statements.Erdos588RichLinesSubquadratic

abbrev Point := EuclideanSpace ℝ (Fin 2)

def lineThrough (p q : Point) : Set Point :=
  {r | Collinear ℝ {p, q, r}}

def CandidateLine (A : Finset Point) (L : Set Point) : Prop :=
  ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ L = lineThrough p q

def Admissible (k : ℕ) (A : Finset Point) : Prop :=
  ∀ ⦃p q⦄, p ∈ A → q ∈ A → p ≠ q →
    Set.ncard ((A : Set Point) ∩ lineThrough p q) ≤ k

noncomputable def richLineCount (k : ℕ) (A : Finset Point) : ℕ :=
  Set.ncard {L : Set Point |
    CandidateLine A L ∧ k ≤ Set.ncard ((A : Set Point) ∩ L)}

noncomputable def richLineMax (k n : ℕ) : ℕ :=
  sSup {m : ℕ | ∃ A : Finset Point,
    A.card = n ∧ Admissible k A ∧ m = richLineCount k A}

abbrev statement : Prop :=
  ∀ k : ℕ, 4 ≤ k →
    Tendsto
      (fun n : ℕ => (richLineMax k n : ℝ) / (n : ℝ) ^ 2)
      atTop (𝓝 0)

theorem target : statement := sorry

end Statements.Erdos588RichLinesSubquadratic
```

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