[#P3032] Asymptotic upper bound for lines containing exactly four points

Problem. Erdős Problem 101: For a finite set $S$ of $n$ points in $\mathbb{R}^2$, let a line be called determined if it is the affine span of two distinct points of $S$. For a positive integer $k$, let $\mathcal{L}_k(S)$ denote the set of determined lines that contain exactly $k$ points of $S$. A set $S$ is said to be 5-noncollinear if no line contains five or more points of $S$. Define $f(n)$ to be the supremum of $|\mathcal{L}_4(S)|$ over all 5-noncollinear sets $S \subset \mathbb{R}^2$ with $|S| = n$. Determine whether $f(n) = o(n^2)$ as $n \to \infty$.

A finite mathematical diagram showing planar points with four-point lines marked. — Planar points with several four-point lines marked.

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Definitions and notation

1Context

This problem originates from discrete geometry and concerns the distribution of points on lines in the plane. The condition that no five points are collinear is necessary to avoid trivial configurations where many lines could each contain exactly four points.

2Problem setup

Definition 1 (For points $p, q \in \mathbb{R}^2$ with $p \neq q$, the affine span of $\{p, q\}$). For points $p, q \in \mathbb{R}^2$ with $p \neq q$, the affine span of $\{p, q\}$ is the unique line in $\mathbb{R}^2$ passing through both $p$ and $q$.

Remark 1. For a finite set $S \subset \mathbb{R}^2$ and a positive integer $k$, the set $\mathcal{L}_k(S)$ consists of all lines that are affine spans of two distinct points of $S$ and that contain exactly $k$ points of $S$.

Definition 2 (A set $S \subset \mathbb{R}^2$). A set $S \subset \mathbb{R}^2$ is 5-noncollinear if no line in the plane contains five or more points of $S$.

Remark 2. For functions $g, h: \mathbb{N} \to \mathbb{R}$, we write $g(n) = o(h(n))$ as $n \to \infty$ if $\lim_{n \to \infty} \frac{g(n)}{h(n)} = 0$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 101: For a finite set $S$ of $n$ points in $\mathbb{R}^2$, let a line be called determined if it is the affine span of two distinct points of $S$. For a positive integer $k$, let $\mathcal{L}_k(S)$ denote the set of determined lines that contain exactly $k$ points of $S$. A set $S$ is said to be 5-noncollinear if no line contains five or more points of $S$. Define $f(n)$ to be the supremum of $|\mathcal{L}_4(S)|$ over all 5-noncollinear sets $S \subset \mathbb{R}^2$ with $|S| = n$. Determine whether $f(n) = o(n^2)$ as $n \to \infty$.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 101 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 101: For a finite set $S$ of $n$ points in $\mathbb{R}^2$, let a line be called determined if it is the affine span of two distinct points of $S$. For a positive integer $k$, let $\mathcal{L}_k(S)$ denote the set of determined lines that contain exactly $k$ points of $S$. A set $S$ is said to be 5-noncollinear if no line contains five or more points of $S$. Define $f(n)$ to be the supremum of $|\mathcal{L}_4(S)|$ over all 5-noncollinear sets $S \subset \mathbb{R}^2$ with $|S| = n$. Determine whether $f(n) = o(n^2)$ as $n \to \infty$.^[1]

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 101 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 101: For a finite set $S$ of $n$ points in $\mathbb{R}^2$, let a line be called determined if it is the affine span of two distinct points of $S$. For a positive integer $k$, let $\mathcal{L}_k(S)$ denote the set of determined lines that contain exactly $k$ points of $S$. A set $S$ is said to be 5-noncollinear if no line contains five or more points of $S$. Define $f(n)$ to be the supremum of $\|\mathcal{L}_4(S)\|$ over all 5-noncollinear sets $S \subset \mathbb{R}^2$ with $\|S\| = n$. Determine whether $f(n) = o(n^2)$ as $n \to \infty$.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 101 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 101: For a finite set $S$ of $n$ points in $\mathbb{R}^2$, let a line be called determined if it is the affine span of two distinct points of $S$. For a positive integer $k$, let $\mathcal{L}_k(S)$ denote the set of determined lines that contain exactly $k$ points of $S$. A set $S$ is said to be 5-noncollinear if no line contains five or more points of $S$. Define $f(n)$ to be the supremum of $\|\mathcal{L}_4(S)\|$ over all 5-noncollinear sets $S \subset \mathbb{R}^2$ with $\|S\| = n$. Determine whether $f(n) = o(n^2)$ as $n \to \infty$.^[1] Relevance to this problem For erdos problem 101, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 101 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 101. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 101 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 101: For a finite set $S$ of $n$ points in $\mathbb{R}^2$, let a line be called determined if it is the affine span of two distinct points of $S$. For a positive integer $k$, let $\mathcal{L}_k(S)$ denote the set of determined lines that contain exactly $k$ points of $S$. A set $S$ is said to be 5-noncollinear if no line contains five or more points of $S$. Define $f(n)$ to be the supremum of $\|\mathcal{L}_4(S)\|$ over all 5-noncollinear sets $S \subset \mathbb{R}^2$ with $\|S\| = n$. Determine whether $f(n) = o(n^2)$ as $n \to \infty$. Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 101: For a finite set $S$ of $n$ points in $\mathbb{R}^2$, let a line be called determined if it is the affine span of two distinct points of $S$. For a positive integer $k$, let $\mathcal{L}_k(S)$ denote the set of determined lines that contain exactly $k$ points of $S$. A set $S$ is said to be 5-noncollinear if no line contains five or more points of $S$. Define $f(n)$ to be the supremum of $\|\mathcal{L}_4(S)\|$ over all 5-noncollinear sets $S \subset \mathbb{R}^2$ with $\|S\| = n$. Determine whether $f(n) = o(n^2)$ as $n \to \infty$.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 101: For a finite set $S$ of $n$ points in $\mathbb{R}^2$, let a line be called determined if it is the affine span of two distinct points of $S$. For a positive integer $k$, let $\mathcal{L}_k(S)$ denote the set of determined lines that contain exactly $k$ points of $S$. A set $S$ is said to be 5-noncollinear if no line contains five or more points of $S$. Define $f(n)$ to be the supremum of $\|\mathcal{L}_4(S)\|$ over all 5-noncollinear sets $S \subset \mathbb{R}^2$ with $\|S\| = n$. Determine whether $f(n) = o(n^2)$ as $n \to \infty$.^[1] Relevance to this problem For Asymptotic upper bound for lines containing exactly four points, record erdos-problem-101-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 101: For a finite set $S$ of $n$ points in $\mathbb{R}^2$, let a line be called determined if it is the affine span of two distinct points of $S$. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 101: For a finite set $S$ of $n$ points in $\mathbb{R}^2$, let a line be called determined if it is the affine span of two distinct points of $S$. For a positive integer $k$, let $\mathcal{L}_k(S)$ denote the set of determined lines that contain exactly $k$ points of $S$. A set $S$ is said to be 5-noncollinear if no line contains five or more points of $S$. Define $f(n)$ to be the supremum of $\|\mathcal{L}_4(S)\|$ over all 5-noncollinear sets $S \subset \mathbb{R}^2$ with $\|S\| = n$. Determine whether $f(n) = o(n^2)$ as $n \to \infty$. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 101 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 101 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

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The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemAsymptotic upper bound for lines containing exactly four points

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Grid points with no three collinear and no four concyclicdiscrete geometry
Largest equilateral-triangle-free subset of the fifteen-row triangular latticediscrete geometry
Most lattice points with all pairwise slopes distinct in a ten by ten griddiscrete geometry

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“Asymptotic upper bound for lines containing exactly four points.” TheoremDB. P3032. Problem statement; statement text SHA-256 67bb3dc7a2dc210f335ad972aab324ff13f3b6ef32e2b2fb92453f855b22c1b7. https://theoremdb.org/statement/?ref=P3032

BibTeX

@misc{theoremdb-problem-67bb3dc7a2dc210f335ad972aab324ff13f3b6ef32e2b2fb92453f855b22c1b7,
  title = {{Asymptotic upper bound for lines containing exactly four points}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 67bb3dc7a2dc210f335ad972aab324ff13f3b6ef32e2b2fb92453f855b22c1b7},
  url = {https://theoremdb.org/statement/?ref=P3032}
}

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This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

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1References

Packet source. Erdős Problems database, Problem 101, maintained status record. Erdős Problems record 101, checked 2026-08-01. Problem 101; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 101; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Asymptotic upper bound for lines containing exactly four points: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 101 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Asymptotic upper bound for lines containing exactly four points: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 101. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/101.lean:L51; theorem erdos_101; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Asymptotic upper bound for lines containing exactly four points: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

[#P3032] Asymptotic upper bound for lines containing exactly four points

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