[#P3044] Asymptotic growth of consecutive van der Waerden number quotients

A finite mathematical diagram showing two-colored intervals with monochromatic arithmetic progressions. — Two-colored intervals with arithmetic progressions marked.

Problem. Erdős Problem 138: For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$. Does the sequence of quotients $\frac{W(k+1)}{W(k)}$ tend to infinity as $k \to \infty$? In other words, does $\displaystyle\lim_{k \to \infty} \frac{W(k+1)}{W(k)} = \infty$ hold?

1Context

This problem is a variant of Erdős Problem 138 concerning the growth rate of van der Waerden numbers. The van der Waerden number $W(k)$ is known to exist for all $k$ by van der Waerden's theorem. Related results include Berlekamp's lower bound $W(p+1) \geq p \cdot 2^p$ for prime $p$, Gowers' upper bound, and the resolved question of whether $W(k+1) - W(k) \to \infty$.

2Problem setup

Definition 1 (A $r$-coloring of a set $S$). A $r$-coloring of a set $S$ is a function from $S$ to $\{1, \ldots, r\}$, assigning one of $r$ colors to each element of $S$.

Definition 2 (A monochromatic arithmetic progression of length $k$ in a coloring). A monochromatic arithmetic progression of length $k$ in a coloring is a set of $k$ equally spaced integers that all receive the same color.

Definition 3 (The van der Waerden number $W(k)$ for $2$ colors). The van der Waerden number $W(k)$ for $2$ colors is the smallest positive integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$.

Remark 1. This problem is a variant of Erdős Problem 138 concerning the growth rate of van der Waerden numbers. The van der Waerden number $W(k)$ is known to exist for all $k$ by van der Waerden's theorem. Related results include Berlekamp's lower bound $W(p+1) \geq p \cdot 2^p$ for prime $p$, Gowers' upper bound, and the resolved question of whether $W(k+1) - W(k) \to \infty$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 138: For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$. Does the sequence of quotients $\frac{W(k+1)}{W(k)}$ tend to infinity as $k \to \infty$? In other words, does $\displaystyle\lim_{k \to \infty} \frac{W(k+1)}{W(k)} = \infty$ hold?

1Status

Current status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 138 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 138: For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$. Does the sequence of quotients $\frac{W(k+1)}{W(k)}$ tend to infinity as $k \to \infty$? In other words, does $\displaystyle\lim_{k \to \infty} \frac{W(k+1)}{W(k)} = \infty$ hold?^[1]

1Packet records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 138 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 138: For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$. Does the sequence of quotients $\frac{W(k+1)}{W(k)}$ tend to infinity as $k \to \infty$? In other words, does $\displaystyle\lim_{k \to \infty} \frac{W(k+1)}{W(k)} = \infty$ hold?^[1] Relevance to this problem For erdos problem 138, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 138 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 138. 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 138 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 138: For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$. Does the sequence of quotients $\frac{W(k+1)}{W(k)}$ tend to infinity as $k \to \infty$? In other words, does $\displaystyle\lim_{k \to \infty} \frac{W(k+1)}{W(k)} = \infty$ hold? Open full record →
	Trace	Reported
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 138: For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$. Does the sequence of quotients $\frac{W(k+1)}{W(k)}$ tend to infinity as $k \to \infty$? In other words, does $\displaystyle\lim_{k \to \infty} \frac{W(k+1)}{W(k)} = \infty$ hold?^[1] Relevance to this problem For Asymptotic growth of consecutive van der Waerden number quotients, record erdos-problem-138-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 138: For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$. 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 138: For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$. Does the sequence of quotients $\frac{W(k+1)}{W(k)}$ tend to infinity as $k \to \infty$? In other words, does $\displaystyle\lim_{k \to \infty} \frac{W(k+1)}{W(k)} = \infty$ hold? 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 materialContext, examples, and computations

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 138 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 138 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 connectTyped relations and evidence flow

ProblemAsymptotic growth of consecutive van der Waerden number quotients

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

All nonnegative limits of normalized consecutive-prime gapsnumber theory
Decidability of positivity for linear recurrence sequencesnumber theory
Erdős Problem 152: isolated sums in Sidon setsnumber theory

How to cite

TheoremDB contributors, “Asymptotic growth of consecutive van der Waerden number quotients,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-138

This page as plain text: erdos-problem-138.md

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 138, maintained status record. Erdős Problems record 138, checked 2026-08-01. Problem 138; 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 138; 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 growth of consecutive van der Waerden number quotients: 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 138 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 growth of consecutive van der Waerden number quotients: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 138. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/138.lean:L106; theorem erdos_138.variants.quotient; 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 growth of consecutive van der Waerden number quotients: 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.

[#P3044] Asymptotic growth of consecutive van der Waerden number quotients

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance