[#P3048] Asymptotic formula for the maximum size of a 3-term-AP-free subset of \(\{1,\dots,N\}\)

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Problem. 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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Definitions and notation

1Context

This problem concerns the quantitative behavior of sets avoiding 3-term arithmetic progressions, a central topic in additive combinatorics. The function \(r_3(N)\) measures how large a subset of the first \(N\) positive integers can be while avoiding any three distinct elements in arithmetic progression.

2Definitions

Definition 1 (For positive integers \(k\) and \(N\), the quantity \(r_k(N)\). For positive integers \(k\) and \(N\), the quantity \(r_k(N)\) is defined as the maximum cardinality of a subset \(A \subseteq \{1, 2, \dots, N\}\) such that \(A\) contains no non-trivial \(k\)-term arithmetic progression. A \(k\)-term arithmetic progression is a sequence of the form \(a, a+d, a+2d, \dots, a+(k-1)d\), and it is called non-trivial when the common difference \(d\) is non-zero, or equivalently when all \(k\) terms are distinct.

Definition 2 (Definition 2). For functions \(f, g: \mathbb{N} \to \mathbb{R}_{\geq 0}\), we write \(f(N) = \Theta(g(N))\) as \(N \to \infty\) if there exist positive constants \(c_1, c_2\) and a positive integer \(N_0\) such that \(c_1 g(N) \leq f(N) \leq c_2 g(N)\) for all \(N \geq N_0\).

3What counts as a solution

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.

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 142 as open. The unresolved remainder is the full displayed statement. 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.^[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 142 as open. The unresolved remainder is the full displayed statement. 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.	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 142 as open. The unresolved remainder is the full displayed statement. 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.^[1] Relevance to this problem For erdos problem 142, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 142 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 142. 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 142 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 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. Open full record →
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.	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 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.^[1] Relevance to this problem 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 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 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. 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 142 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 142 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

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

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Asymptotic formula for the maximum size of a 3-term-AP-free subset of \(\{1,\dots,N\}\).” TheoremDB. P3048. Problem statement; statement text SHA-256 1617e03b4c24d5ac55a92da770870f52986c5c54c03ceefdeb9c11e792b0dae7. https://theoremdb.org/statement/?ref=P3048

BibTeX

@misc{theoremdb-problem-1617e03b4c24d5ac55a92da770870f52986c5c54c03ceefdeb9c11e792b0dae7,
  title = {{Asymptotic formula for the maximum size of a 3-term-AP-free subset of \textbackslash{}(\textbackslash{}\{1,\textbackslash{}dots,N\textbackslash{}\}\textbackslash{})}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 1617e03b4c24d5ac55a92da770870f52986c5c54c03ceefdeb9c11e792b0dae7},
  url = {https://theoremdb.org/statement/?ref=P3048}
}

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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.

1References

Packet source. Erdős Problems database, Problem 142, maintained status record. Erdős Problems record 142, checked 2026-08-01. Problem 142; 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 142; 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 formula for the maximum size of a 3-term-AP-free subset of \(\{1,\dots,N\}\): 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 142 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 formula for the maximum size of a 3-term-AP-free subset of \(\{1,\dots,N\}\): Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 142. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/142.lean:L67; theorem erdos_142.variants.three; 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 formula for the maximum size of a 3-term-AP-free subset of \(\{1,\dots,N\}\): 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.

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[#P3048] Asymptotic formula for the maximum size of a 3-term-AP-free subset of \(\{1,\dots,N\}\)

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