# P3044: Asymptotic growth of consecutive van der Waerden number quotients

- ID: `P3044`
- Reference: `erdos-problem-138`
- Page: https://theoremdb.org/statements/P3044
- Record maturity: Reviewed problem with recorded work

## 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?

### Context

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 $W(k+1) - W(k) \to \infty$.

### Problem setup

- **Definition (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 (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 (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.** 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 $W(k+1) - W(k) \to \infty$.

### What 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?

## 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. 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](#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 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?

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?

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-138`, 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 138, maintained status record. Erdős Problems record 138, checked 2026-08-01. Problem 138; status field and linked bibliography https://www.erdosproblems.com/138
   - 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
   - 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 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 138 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 Asymptotic growth of consecutive van der Waerden number quotients: 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 138. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/138.lean:L106; theorem erdos_138.variants.quotient; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/138.lean#L106
   - 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 Asymptotic growth of consecutive van der Waerden number quotients: Supplies the pinned formal declaration whose human-readable editorial statement is published here.