# P3032: Asymptotic upper bound for lines containing exactly four points

- ID: `P3032`
- Reference: `erdos-problem-101`
- Page: https://theoremdb.org/statements/P3032
- Record maturity: Reviewed problem with recorded work

## 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 $n \to \infty$.

### Context

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.

### Problem setup

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

### What 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 $n \to \infty$.

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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 $n \to \infty$.

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 $n \to \infty$.

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-101`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 101, maintained status record. Erdős Problems record 101, checked 2026-08-01. Problem 101; status field and linked bibliography https://www.erdosproblems.com/101
   - 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
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 101 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source 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.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 101. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/101.lean:L51; theorem erdos_101; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/101.lean#L51
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source 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.