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Math-2225 · mathematics · Combinatorics

Exact van der Waerden numbers

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $W(r,k)$ be the least $N$ such that every coloring of $\{1,\ldots,N\}$ with $r$ colors contains a monochromatic arithmetic progression of length $k$. Determine the unknown exact values of $W(r,k)$.

Formulation references: [S1], [S2].

Short English statement

Let $W(r,k)$ be the least $N$ such that every coloring of $\{1,\ldots,N\}$ with $r$ colors contains a monochromatic arithmetic progression of length $k$. Determine the unknown exact values of $W(r,k)$.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2225-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2225

Superexponential van der Waerden growth: The theorem gives a uniform superexponential lower bound for all large progression lengths, but no exact values of W(r,k). It improves extremal bounds within the same family rather than completing the exact-value request.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Exact van der Waerden numbers

OpenAI's claim

Superexponential van der Waerden growth: The claimed lower bound is $W_r(k)>k^{ck\lfloor\log_2 r\rfloor}$ for one absolute positive $c$ once $k$ is large enough. In particular, for each fixed color count at least two, the $k$th root of $W_r(k)$ is unbounded.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $W_r(k)$ be the least $N$ such that every $r$-coloring of $\{1,\ldots,N\}$ has a monochromatic nonconstant $k$-term arithmetic progression. The claimed lower bound is \[W_r(k)>k^{10^{-5}k\lfloor\log_2r\rfloor}\quad(k\ge K_0,\ r\ge2),\] where $K_0$ is absolute. The quantifier is uniform over all integer color counts $r\ge2$, but the estimate does not give exact van der Waerden numbers.

Outline of the argument

Superexponential van der Waerden growth: A cyclic-group coloring uses two circle-coordinate meshes related by an arithmetically chosen dilation. Counting anchored progression patterns gives a balanced-color versus affine-coordinate alternative uniformly in the growing dimension. A norm-dependent perturbation excludes the affine alternative, after which interval and product-color transfers give the stated bounds for all color counts.

Formal verification

Family 160 describes Lean coverage for the uniform van der Waerden lower bound and its growth-limit consequences.

Superexponential van der Waerden growth: The manuscript links QuantitativeVanDerWaerden.lean (declaration OAI.QuantitativeVanDerWaerden.uniform_lower_bound).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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