ProblemP3048

Asymptotic formula for the maximum size of a 3-term-AP-free subset of \(\{1,\dots,N\}\)

Problem statement

Erdős Problem 142: For a positive integer \(N\), let \(r_3(N)\) denote the largest possible cardinality of a subset of \(\{1, 2, \dots, N\}\) that contains no non-trivial 3-term arithmetic progression. A 3-term arithmetic progression is called non-trivial if its terms are distinct. Determine whether there exists a function \(f: \mathbb{N} \to \mathbb{R}\) such that \(r_3(N) = \Theta(f(N))\) as \(N \to \infty\), and if so, identify such a function. That is, find an explicit asymptotic formula for \(r_3(N)\) up to constant factors.

Recorded solution requirements

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 142: For a positive integer \(N\), let \(r_3(N)\) denote the largest possible cardinality of a subset of \(\{1, 2, \dots, N\}\) that contains no non-trivial 3-term arithmetic progression. A 3-term arithmetic progression is called non-trivial if its terms are distinct. Determine whether there exists a function \(f: \mathbb{N} \to \mathbb{R}\) such that \(r_3(N) = \Theta(f(N))\) as \(N \to \infty\), and if so, identify such a function. That is, find an explicit asymptotic formula for \(r_3(N)\) up to constant factors.

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TheoremDB content license: CC0-1.0
Stable alias: erdos-problem-142-attempt-resolution-route
Source: www.erdosproblems.com ↗

Problem packetResearch packetR1205

R1205Recorded attemptOutcome: Open

Resolve the stated acceptance condition

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Authored summary

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 142: For a positive integer \(N\), let \(r_3(N)\) denote the largest possible cardinality of a subset of \(\{1, 2, \dots, N\}\) that contains no non-trivial 3-term arithmetic progression. A 3-term arithmetic progression is called non-trivial if its terms are distinct. Determine whether there exists a function \(f: \mathbb{N} \to \mathbb{R}\) such that \(r_3(N) = \Theta(f(N))\) as \(N \to \infty\), and if so, identify such a function. That is, find an explicit asymptotic formula for \(r_3(N)\) up to constant factors.

The author reports this result. The outcome applies to this attempt's recorded scope.

Attempt outcome: open strategy

Recorded scope: No scope is recorded.

Originating problem: Asymptotic formula for the maximum size of a 3-term-AP-free subset of \(\{1,\dots,N\}\)

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Authored record and scope

Authored title: Resolve the stated acceptance condition
Record type: attempt
Stored status: open_strategy
Evidence grade: self_reported

Work and source credit

Recorded action: No action description supplied.
Authored result summary: Read complete authored result summary
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 142: For a positive integer \(N\), let \(r_3(N)\) denote the largest possible cardinality of a subset of \(\{1, 2, \dots, N\}\) that contains no non-trivial 3-term arithmetic progression. A 3-term arithmetic progression is called non-trivial if its terms are distinct. Determine whether there exists a function \(f: \mathbb{N} \to \mathbb{R}\) such that \(r_3(N) = \Theta(f(N))\) as \(N \to \infty\), and if so, identify such a function. That is, find an explicit asymptotic formula for \(r_3(N)\) up to constant factors.
Reported outcome: No separate outcome supplied.
Recorded status: open_strategy
Recorded evidence grade: self_reported
Recorded scope: No explicit scope supplied.

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2Authored explanation

Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 142: For a positive integer \(N\), let \(r_3(N)\) denote the largest possible cardinality of a subset of \(\{1, 2, \dots, N\}\) that contains no non-trivial 3-term arithmetic progression. A 3-term arithmetic progression is called non-trivial if its terms are distinct. Determine whether there exists a function \(f: \mathbb{N} \to \mathbb{R}\) such that \(r_3(N) = \Theta(f(N))\) as \(N \to \infty\), and if so, identify such a function. That is, find an explicit asymptotic formula for \(r_3(N)\) up to constant factors. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.

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Agent handoff

Use this record for: For Asymptotic formula for the maximum size of a 3-term-AP-free subset of \(\{1,\dots,N\}\), record erdos-problem-142-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 142: For a positive integer \(N\), let \(r_3(N)\) denote the largest possible cardinality of a subset of \(\{1, 2, \dots, N\}\) that contains no non-trivial 3-term arithmetic progression.
Evidence boundary: The recorded outcome applies to this attempt's stated scope.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
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Verification source: www.erdosproblems.com ↗, Editorial research route recorded 2026-07-31

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Addresses

R1205addressesCurrent status and unresolved remainder
claim

Recorded for

R1205recorded forerdos problem 142
problem

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6Provenance

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Project: erdos-problem-142-source-review
Locator: Editorial research route recorded 2026-07-31
License: CC0-1.0
Contributors: TheoremDB maintainers
Source: www.erdosproblems.com ↗
Public record: R1205
Stable alias: erdos-problem-142-attempt-resolution-route
Projection: Reproduction fields are derived from the immutable record.

A route someone took, recorded so the next person can reuse it or avoid it.

Resolve the stated acceptance condition

Work and source credit

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4How it connects

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