[#P2976] Erdős–Rado sunflower threshold growth rate

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?

A finite mathematical diagram showing finite sets with a common intersection. — Finite sets arranged around their common intersection.

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Definitions and notation

1Context

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.

2Definitions

Definition 1 (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 2 (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.

3What 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?

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 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]

1Packet records

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Record	Kind	Assessment
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?	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 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] Relevance to this problem For erdos problem 20, pins the dated research frontier: 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. 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 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? Open full record →
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?	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 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] Relevance to this problem For Erdős–Rado sunflower threshold growth rate, record erdos-problem-20-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 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). 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 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? 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 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.

How the 2 records connect

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

ProblemErdős–Rado sunflower threshold growth rate

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Constructing a union from non-cancelling intersectionscombinatorics
Plane tilings by every five-cell lattice animalcombinatorics
Cycle Double Cover Conjecturecombinatorics

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Plain text

“Erdős–Rado sunflower threshold growth rate.” TheoremDB. P2976. Problem statement; statement text SHA-256 72afd1d828e9f47bfb49724db532d3d247476af9a1ef02d9a2e0d631355a8037. https://theoremdb.org/statement/?ref=P2976

BibTeX

@misc{theoremdb-problem-72afd1d828e9f47bfb49724db532d3d247476af9a1ef02d9a2e0d631355a8037,
  title = {{Erdős–Rado sunflower threshold growth rate}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 72afd1d828e9f47bfb49724db532d3d247476af9a1ef02d9a2e0d631355a8037},
  url = {https://theoremdb.org/statement/?ref=P2976}
}

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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.

1Lean verification

1References

Packet source. Erdős Problems database, Problem 20, maintained status record. Erdős Problems record 20, checked 2026-08-01. Problem 20; 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 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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 20 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 Erdős–Rado sunflower threshold growth rate: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 20. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/20.lean:L51; theorem erdos_20; 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 Erdős–Rado sunflower threshold growth rate: 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.

[#P2976] Erdős–Rado sunflower threshold growth rate

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