{ "artifactId": "artifact.library.mathlib.hales-jewett-theorem.v001", "candidateOnly": false, "category": "Combinatorics", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "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", "family": "Ramsey theory", "id": "library.mathlib.hales-jewett-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/hales-jewett-theorem/", "entryData": "/data/library-theorems/hales-jewett-theorem.json", "evidence": "/proofs/artifact.library.mathlib.hales-jewett-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.hales-jewett-theorem.v001.evidence.json", "source": "/sources/upstream/hales-jewett-theorem/" }, "landmark": { "completeFormalType": "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", "concepts": [ "Ramsey theory", "finite word hypercubes", "combinatorial lines", "finite colorings", "color focusing", "product argument" ], "plainLanguage": "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.", "poster": { "alt": "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.", "byteSize": 1919823, "caption": "For every finite alphabet and finite color set, some finite coordinate type makes every coloring contain a monochromatic combinatorial line.", "description": "The theorem block preserves the finite alphabet and color hypotheses, existential finite coordinate type, universal coloring, and monochromatic-line conclusion. The central example gives the complete line for alphabet {a,b,c} under the pattern (a,x,x,b,x): (a,a,a,b,a), (a,b,b,b,b), and (a,c,c,b,c), all with one whole-word halo. The footer explicitly records the non-effective and one-dimensional boundary.", "derivatives": [ { "byteSize": 85288, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hales-jewett-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:a31d606cc1468aef250ad90670164656b1ebc986de1ee37ed7fb9950c3118e9d", "width": 640 }, { "byteSize": 154298, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hales-jewett-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:bd07657d0f16dd71fe13dfaaf7886b8d72a22871338333f115c0dc7a835c6790", "width": 1024 } ], "generatedAt": "2026-07-25T16:41:46-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/hales-jewett-theorem/theorem-poster-v1.png", "sha256": "sha256:65f8b4480c59b6a9f9f498ea7301f3391043abc10a4352a513209b06f37e53ba", "title": "Hales–Jewett Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "RAMSEY THEORY · WORD HYPERCUBES\nHALES–JEWETT THEOREM\nFOR EVERY FINITE ALPHABET α AND FINITE COLOR SET κ,\nTHERE EXISTS A FINITE COORDINATE TYPE ι.\nEVERY COLORING C : (ι → α) → κ\nHAS A MONOCHROMATIC COMBINATORIAL LINE.\nFIX SOME COORDINATES · VARY A NONEMPTY SET TOGETHER\nEXACT SCOPE\nEXISTENCE ONLY: NO EFFECTIVE DIMENSION BOUND.\nTHIS IS THE LINE THEOREM, NOT THE MULTIDIMENSIONAL VERSION." }, "visual": { "alt": "Three five-symbol words share one whole-word color; positions two, three, and five change together while positions one and four stay fixed.", "byteSize": 2438519, "caption": "In a sufficiently large finite word space, every finite coloring contains a line formed by varying one nonempty coordinate set together.", "description": "An abstract colored lattice represents the surrounding word hypercube. In front, the complete three-point line for a three-symbol alphabet has pattern (triangle, x, x, square, x): the triangle and square coordinates stay fixed, the other three positions vary synchronously, and one common vermilion outline marks the coloring of every whole word.", "derivatives": [ { "byteSize": 43062, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hales-jewett-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:3b30d64518424d02f062b612b6456516a097283436bc354aa40a0aac9db761cd", "width": 640 }, { "byteSize": 114430, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hales-jewett-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:bffa737fd8c1d9fe2c77fdb7bebfc1a5e8e4c69bb2a9abb8103e956908f00d1d", "width": 1200 } ], "generatedAt": "2026-07-25T16:41:46-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/hales-jewett-theorem/theorem-schematic-v1.png", "sha256": "sha256:5c871aafda13b1242aece7e758ac85836593d89eb0c9013b084ec5ab7f2da174", "title": "A monochromatic line through a colored word hypercube", "width": 1536 }, "walkthrough": null, "whyLandmark": "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." }, "nonClaims": [ "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." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.hales-jewett-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "For every finite alphabet and finite color set, some finite word dimension makes every coloring contain a monochromatic combinatorial line.", "targetId": "target.library.mathlib.hales-jewett-theorem.v001", "title": "Hales–Jewett Theorem", "upstreamOrigin": { "declarationName": "Combinatorics.Line.exists_mono_in_high_dimension", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Combinatorics/HalesJewett.lean#L448", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:da5701045beed71be46096d789c63c5a952421927b0e1407b625403392449db4", "sourceByteLength": 25839, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Combinatorics/HalesJewett.lean", "sourceLine": 448, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }