Combinatorics · existing Mathlib declaration

Hales–Jewett Theorem

Fix a finite alphabet and finitely many colors. In some sufficiently large finite word space, no matter how every word is colored, there is a family of words obtained by changing one nonempty set of positions together, all with the same color.

Ramsey theory
finite word hypercubes
combinatorial lines
finite colorings
color focusing
product argument

The theorem at a glance

Hales–Jewett Theorem at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

A portrait theorem poster states the finite-alphabet Hales–Jewett theorem and shows the complete three-word combinatorial line for the alphabet a, b, c. — For every finite alphabet and finite color set, some finite coordinate type makes every coloring contain a monochromatic combinatorial line.

Formal orientation

Statement map

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

Selected after rejecting a brace that incorrectly included fixed coordinate 4 and after correcting the evaluation alphabet. The left panel orders finite α,κ; existential finite ι; universal C; existential line and color; then universal x. The center has exactly five coordinates, fixes 1 and 4, and connects only 2, 3, and 5 to one shared wildcard x. The right table gives the complete {a,b,c} line and one shared color c. The boundary says one-dimensional, existence only, and no effective ι bound. — In a sufficiently large finite word space, every finite coloring contains a line formed by varying one nonempty coordinate set together.

Complementary intuition

A mathematical landmark

The Hales–Jewett theorem is a central structural result in Ramsey theory. Mathlib’s source exposes the theorem’s distinctive route through induction on the alphabet, color-focused families, a product argument, and the final impossibility of more distinct focused colors than the finite color type contains.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

theorem Combinatorics.Line.exists_mono_in_high_dimension (α : Type u) [Finite α] (κ : Type v) [Finite κ] : ∃ (ι : Type) (_ : Fintype ι), ∀ C : (ι → α) → κ, ∃ l : Combinatorics.Line α ι, l.IsMono C

Selected evidence declaration text

theorem Combinatorics.Line.exists_mono_in_high_dimension (α : Type u) [Finite α] (κ : Type v) [Finite κ] : ∃ (ι : Type) (_ : Fintype ι), ∀ C : (ι → α) → κ, ∃ l : Combinatorics.Line α ι, l.IsMono C

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Claim boundary

No new theorem is claimed

This page indexes Mathlib’s combinatorial-line form of the Hales–Jewett theorem for arbitrary finite types α and κ. It produces some finite coordinate type ι before quantifying over every coloring C : (ι → α) → κ. A line fixes some coordinates and varies one nonempty set of coordinates together through a common letter. The selected declaration supplies no effective or least dimension, canonical choice, coloring-independent line, or algorithm, and it is not the separate multidimensional subspace theorem or the Van der Waerden corollary.

ProofAtlas did not originate the Hales–Jewett theorem or Mathlib’s declaration.
The selected declaration gives no effective upper bound or least value for the finite coordinate dimension.
The displayed five-coordinate line is an illustration, not a universally sufficient Hales–Jewett dimension.
The selected declaration is not the separate multidimensional Hales–Jewett theorem and does not itself state Van der Waerden’s theorem.
The generated explanation and visuals are not proof evidence.

Provenance

Origin and evidence stay separate

Existing declaration: Combinatorics.Line.exists_mono_in_high_dimension in mathlib
Relationship: The checked artifact is the upstream declaration itself
Preferred checked artifact: artifact.library.mathlib.hales-jewett-theorem.v001
Source handling: Verified upstream reference; no mirrored source package