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Math-1577 · mathematics · Combinatorics

An explicit local-lemma certificate and bounded high-overlap neighbourhood for van der Waerden hypergraphs

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

For an integer $k\ge2$, a nonconstant $k$-term arithmetic progression in $[N]=\{1,\ldots,N\}$ is a set \[ \{a,a+d,\ldots,a+(k-1)d\}, \qquad a,d\in\mathbb Z,\quad d\ge1, \] whose terms all lie in $[N]$. A two-coloring is any map $c:[N]\to\{\mathrm{red},\mathrm{blue}\}$. A progression is monochromatic when $c$ has the same value on all its terms. Let $W(k)$ be the least $N$ for which every two-coloring of $[N]$ contains a monochromatic $k$-term progression. The van der Waerden theorem ensures that $W(k)$ is finite.

The workshop asks for the true order of growth of $W(k)$ as $k\to\infty$. It singles out the proposed upper bound \[ W(k)\le2^{k^2} \qquad(k\ge2). \] The exponent is $k^2$; the expression is not $2k^2$, as a flattened text extraction could suggest. The two-color convention is fixed throughout and is not an additional growing parameter. An upper bound must apply to every coloring of the entire interval, while a lower bound is provided by a coloring of a shorter interval avoiding all such monochromatic progressions. A coloring that avoids only progressions with some specified differences does not suffice. The broad growth question remains distinct from the particular $2^{k^2}$ target: settling the latter would not itself determine the sharp asymptotic order.

Short English statement

Determine the growth of the two-color van der Waerden numbers, including the proposed upper bound two to the power k squared.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1577-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1577

The claimed estimate improves the lower bound for the two-color diagonal quantity $W(k)$ to $k^{10^{-5}k}$ once $k$ is sufficiently large. It leaves both the sharp growth rate and the proposed upper bound $W(k)\le2^{k^2}$ undetermined.

An explicit local-lemma certificate and bounded high-overlap neighbourhood for van der Waerden hypergraphs

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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