# P2976: Erdős–Rado sunflower threshold growth rate

- ID: `P2976`
- Reference: `erdos-problem-20`
- Page: https://theoremdb.org/statements/P2976
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 20: For positive integers $n$ and $k$, let $f(n,k)$ denote the smallest integer such that every family of $n$-element sets with at least $f(n,k)$ members contains $k$ sets whose pairwise intersections are all equal (such a configuration is called a $k$-sunflower). Does there exist a function $c \colon \mathbb{N} \to \mathbb{N}$ such that for all positive integers $n$ and all positive integers $k$, the inequality $f(n,k) < (c(k))^n$ holds?

### Context

This problem concerns the growth rate of the sunflower threshold function introduced by Erdős and Rado. The best known general upper bound is $f(n,k) \leq (k-1)^n \cdot n! + 1$, which grows faster than any exponential function in $n$ with base depending only on $k$. The question asks whether this factorial bound can be improved to a pure exponential bound.

### Problem setup

- **Definition (A family of sets $\mathcal{S}$).** A family of sets $\mathcal{S}$ is called a $k$-sunflower if it consists of $k$ distinct sets such that the intersection of any two distinct sets in $\mathcal{S}$ is the same fixed set, called the core of the sunflower.
- **Definition (For positive integers $n$ and $k$, the sunflower threshold $f(n,k)$).** For positive integers $n$ and $k$, the sunflower threshold $f(n,k)$ is defined as the minimum integer $m$ such that every family $\mathcal{F}$ of sets, each containing exactly $n$ elements, with $|\mathcal{F}| \geq m$ contains some subfamily $\mathcal{S} \subseteq \mathcal{F}$ with $|\mathcal{S}| = k$ that forms a $k$-sunflower.
- **Remark.** This problem concerns the growth rate of the sunflower threshold function introduced by Erdős and Rado. The best known general upper bound is $f(n,k) \leq (k-1)^n \cdot n! + 1$, which grows faster than any exponential function in $n$ with base depending only on $k$. The question asks whether this factorial bound can be improved to a pure exponential bound.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 20: For positive integers $n$ and $k$, let $f(n,k)$ denote the smallest integer such that every family of $n$-element sets with at least $f(n,k)$ members contains $k$ sets whose pairwise intersections are all equal (such a configuration is called a $k$-sunflower). Does there exist a function $c \colon \mathbb{N} \to \mathbb{N}$ such that for all positive integers $n$ and all positive integers $k$, the inequality $f(n,k) < (c(k))^n$ holds?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 20 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 20: For positive integers $n$ and $k$, let $f(n,k)$ denote the smallest integer such that every family of $n$-element sets with at least $f(n,k)$ members contains $k$ sets whose pairwise intersections are all equal (such a configuration is called a $k$-sunflower). Does there exist a function $c \colon \mathbb{N} \to \mathbb{N}$ such that for all positive integers $n$ and all positive integers $k$, the inequality $f(n,k) < (c(k))^n$ holds? [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 20 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 20: For positive integers $n$ and $k$, let $f(n,k)$ denote the smallest integer such that every family of $n$-element sets with at least $f(n,k)$ members contains $k$ sets whose pairwise intersections are all equal (such a configuration is called a $k$-sunflower). Does there exist a function $c \colon \mathbb{N} \to \mathbb{N}$ such that for all positive integers $n$ and all positive integers $k$, the inequality $f(n,k) < (c(k))^n$ holds?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 20 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 20: For positive integers $n$ and $k$, let $f(n,k)$ denote the smallest integer such that every family of $n$-element sets with at least $f(n,k)$ members contains $k$ sets whose pairwise intersections are all equal (such a configuration is called a $k$-sunflower). Does there exist a function $c \colon \mathbb{N} \to \mathbb{N}$ such that for all positive integers $n$ and all positive integers $k$, the inequality $f(n,k) < (c(k))^n$ holds?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 20 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 20 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 20: For positive integers $n$ and $k$, let $f(n,k)$ denote the smallest integer such that every family of $n$-element sets with at least $f(n,k)$ members contains $k$ sets whose pairwise intersections are all equal (such a configuration is called a $k$-sunflower). Does there exist a function $c \colon \mathbb{N} \to \mathbb{N}$ such that for all positive integers $n$ and all positive integers $k$, the inequality $f(n,k) < (c(k))^n$ holds? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-20`, 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 20, maintained status record. Erdős Problems record 20, checked 2026-08-01. Problem 20; status field and linked bibliography https://www.erdosproblems.com/20
   - Also cited at Problem 20; 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 Erdős–Rado sunflower threshold growth rate: 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 20 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 Erdős–Rado sunflower threshold growth rate: 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 20. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/20.lean:L51; theorem erdos_20; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/20.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 Erdős–Rado sunflower threshold growth rate: Supplies the pinned formal declaration whose human-readable editorial statement is published here.