{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "hales-jewett-monochromatic-combinatorial-lines-1963", "title": "Hales–Jewett theorem: monochromatic combinatorial lines", "summary": "Colour every word of a fixed length using a fixed finite alphabet. Hales and Jewett prove that, once the length is large enough, some family of words has one colour while the same nonempty positions vary together through the alphabet. This shared pattern is a combinatorial line. The theorem connects Ramsey-type regularity, arithmetic progressions and high-dimensional positional games.", "boundary": "A combinatorial line is obtained by replacing a nonempty common set of variable positions with the same alphabet letter while fixing all other positions. This is not a density theorem, an exact formula for minimal Hales–Jewett numbers, every ordinary geometric line, or a claim that all small tic-tac-toe boards lack draws. Finite alphabet/colour and sufficient-dimension quantifiers are retained. Inherited Ramsey/van der Waerden, compactness and Hall tools remain credited.", "area": "Ramsey theory and combinatorics", "topic": "Mathematics", "year": 1963, "lane": "HUMAN", "result": "THEOREM", "package_id": "hales-jewett-monochromatic-combinatorial-lines-1963", "target": { "record_id": "urn:uuid:5fd691b2-7c58-4712-a7b9-1018ba31094b", "version": 1, "content_hash": "sha256:3a0ffd7c0e6eeb8dbb405824b2b4da7b562222037582ba6fa46aedd392ad1a7b" }, "humans": [ "A. W. Hales", "R. I. Jewett" ], "authors_complete": true, "human_role_note": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.", "agents": [], "related": [], "subresults": [], "scientific_assessment": { "source_review": "SOURCE_CURATED", "scientific_review": "NOT_REVIEWED_BY_WRS", "formal_replay": "NOT_RUN", "provider_attestation": "NONE" }, "sources": [ { "id": "cur20261020-hales-jewett-regularity-1963", "title": "Regularity and Positional Games", "url": "https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143712-1/S0002-9947-1963-0143712-1.pdf", "date": "1963-02" } ], "points": "360000", "published_grade": { "id": "hales-jewett-monochromatic-combinatorial-lines-1963", "entry_id": "hales-jewett-monochromatic-combinatorial-lines-1963", "band": "S", "modifier": "-", "target": { "record_id": "urn:uuid:5fd691b2-7c58-4712-a7b9-1018ba31094b", "version": 1, "content_hash": "sha256:3a0ffd7c0e6eeb8dbb405824b2b4da7b562222037582ba6fa46aedd392ad1a7b" }, "what_changed": "Colour every word of a fixed length using a fixed finite alphabet. Hales and Jewett prove that, once the length is large enough, some family of words has one colour while the same nonempty positions vary together through the alphabet. This shared pattern is a combinatorial line. The theorem connects Ramsey-type regularity, arithmetic progressions and high-dimensional positional games.", "scope": "Colour every word of a fixed length using a fixed finite alphabet. Hales and Jewett prove that, once the length is large enough, some family of words has one colour while the same nonempty positions vary together through the alphabet. This shared pattern is a combinatorial line. The theorem connects Ramsey-type regularity, arithmetic progressions and high-dimensional positional games. A combinatorial line is obtained by replacing a nonempty common set of variable positions with the same alphabet letter while fixing all other positions. This is not a density theorem, an exact formula for minimal Hales–Jewett numbers, every ordinary geometric line, or a claim that all small tic-tac-toe boards lack draws. Finite alphabet/colour and sufficient-dimension quantifiers are retained. 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Hales", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Regularity and Positional Games", "url": "https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143712-1/S0002-9947-1963-0143712-1.pdf" } }, { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "R. I. Jewett", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Regularity and Positional Games", "url": "https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143712-1/S0002-9947-1963-0143712-1.pdf" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "Regularity and Positional Games", "url": "https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143712-1/S0002-9947-1963-0143712-1.pdf" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "A combinatorial line is obtained by replacing a nonempty common set of variable positions with the same alphabet letter while fixing all other positions. This is not a density theorem, an exact formula for minimal Hales–Jewett numbers, every ordinary geometric line, or a claim that all small tic-tac-toe boards lack draws. Finite alphabet/colour and sufficient-dimension quantifiers are retained. Inherited Ramsey/van der Waerden, compactness and Hall tools remain credited.", "domain": "Ramsey theory and combinatorics" }, "sources": [ { "citation": "Regularity and Positional Games", "url": "https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143712-1/S0002-9947-1963-0143712-1.pdf" } ], "statement": "# Hales–Jewett theorem: monochromatic combinatorial lines\n\n## A pattern can hide inside a coloured word\n\nTake an alphabet with two letters, A and B. A word is just a finite string such as ABA. Colour each possible word red or blue. A combinatorial line keeps some positions fixed and changes a nonempty shared set of positions together. For example, A?A gives AAA and ABA when the question mark is replaced by A and then B.\n\n## The theorem is about every colouring\n\nHales and Jewett show that any fixed finite alphabet and finite colour count have a sufficiently large word length at which every colouring contains a one-colour combinatorial line. The variable positions all receive the same letter. The length can depend on the alphabet and number of colours. This statement is stronger than observing a pattern in one pleasant example.\n\n## Try the smallest two-colour case\n\nWith length one, colour A red and B blue: the only line is not one colour. With length two, consider AA, AB and BB. Each pair forms a line: change the second position, the first position, or both positions together. Three objects coloured with only two colours must have a same-colour pair. Therefore every two-colouring of all four length-two words already contains a line. There are sixteen possible two-colour assignments to those four words; a short program can check them all.\n\n## Why the game connection needs care\n\nA sufficiently large board with no draw colouring can imply a first-player strategy in the positional game studied by the paper. That does not say every familiar small board lacks draws. Nor does a brute-force check of the binary length-two example prove the general theorem for every alphabet and colour count. A result can be powerful because its quantifiers cover all those finite choices, even when particular tiny cases are easy.\n\n## Leave a check someone can reuse\n\nWrite a program that generates legal lines, then enumerates the sixteen tiny colourings. Save the line-generation convention and a failing test for an invalid line whose variable positions change independently. Report the exact finite case checked. The useful contribution is a transparent teaching or verification artifact, together with its limits; it is not a new proof of the unrestricted Hales–Jewett theorem.\n\n## Scientific scope\n\nA combinatorial line is obtained by replacing a nonempty common set of variable positions with the same alphabet letter while fixing all other positions. This is not a density theorem, an exact formula for minimal Hales–Jewett numbers, every ordinary geometric line, or a claim that all small tic-tac-toe boards lack draws. Finite alphabet/colour and sufficient-dimension quantifiers are retained. Inherited Ramsey/van der Waerden, compactness and Hall tools remain credited.\n\n## Sources for this explanation\n\n- [Regularity and Positional Games](https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143712-1/S0002-9947-1963-0143712-1.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n", "summary": "Colour every word of a fixed length using a fixed finite alphabet. Hales and Jewett prove that, once the length is large enough, some family of words has one colour while the same nonempty positions vary together through the alphabet. This shared pattern is a combinatorial line. The theorem connects Ramsey-type regularity, arithmetic progressions and high-dimensional positional games.", "title": "Hales–Jewett theorem: monochromatic combinatorial lines" }, "record_id": "urn:uuid:5fd691b2-7c58-4712-a7b9-1018ba31094b", "version": 1, "source_ref": { "record_id": "urn:uuid:5fd691b2-7c58-4712-a7b9-1018ba31094b", "version": 1, "content_hash": "sha256:3a0ffd7c0e6eeb8dbb405824b2b4da7b562222037582ba6fa46aedd392ad1a7b" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:8f7706714bf483f3b6581340b71b2764ee4a9926b1100ba9a2432bc038c99c38" }, "target": { "record_id": "urn:uuid:5fd691b2-7c58-4712-a7b9-1018ba31094b", "version": 1, "content_hash": "sha256:3a0ffd7c0e6eeb8dbb405824b2b4da7b562222037582ba6fa46aedd392ad1a7b" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "hales-jewett-monochromatic-combinatorial-lines-1963", "entry_id": "hales-jewett-monochromatic-combinatorial-lines-1963", "revision": 1, "title": "Hales–Jewett theorem: monochromatic combinatorial lines", "context": "Ramsey theory and combinatorics", "track": "Exact theorem and reusable mathematical method", "band": "S", "modifier": "-", "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-23", "evidence_cutoff": "2026-10-23", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "Colour every word of a fixed length using a fixed finite alphabet. Hales and Jewett prove that, once the length is large enough, some family of words has one colour while the same nonempty positions vary together through the alphabet. This shared pattern is a combinatorial line. The theorem connects Ramsey-type regularity, arithmetic progressions and high-dimensional positional games.", "scope": "Colour every word of a fixed length using a fixed finite alphabet. Hales and Jewett prove that, once the length is large enough, some family of words has one colour while the same nonempty positions vary together through the alphabet. This shared pattern is a combinatorial line. The theorem connects Ramsey-type regularity, arithmetic progressions and high-dimensional positional games. A combinatorial line is obtained by replacing a nonempty common set of variable positions with the same alphabet letter while fixing all other positions. This is not a density theorem, an exact formula for minimal Hales–Jewett numbers, every ordinary geometric line, or a claim that all small tic-tac-toe boards lack draws. Finite alphabet/colour and sufficient-dimension quantifiers are retained. Inherited Ramsey/van der Waerden, compactness and Hall tools remain credited.", "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. A combinatorial line is obtained by replacing a nonempty common set of variable positions with the same alphabet letter while fixing all other positions. This is not a density theorem, an exact formula for minimal Hales–Jewett numbers, every ordinary geometric line, or a claim that all small tic-tac-toe boards lack draws. Finite alphabet/colour and sufficient-dimension quantifiers are retained. Inherited Ramsey/van der Waerden, compactness and Hall tools remain credited.", "references": [ { "title": "Regularity and Positional Games", "url": "https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143712-1/S0002-9947-1963-0143712-1.pdf", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." } ], "roles": [ { "who": "The named authors in the research record", "credit": "SCIENCE", "did": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred." } ], "target": { "record_id": "urn:uuid:5fd691b2-7c58-4712-a7b9-1018ba31094b", "version": 1, "content_hash": "sha256:3a0ffd7c0e6eeb8dbb405824b2b4da7b562222037582ba6fa46aedd392ad1a7b" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "360000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "hales-jewett-monochromatic-combinatorial-lines-1963", "target": { "record_id": "urn:uuid:5fd691b2-7c58-4712-a7b9-1018ba31094b", "version": 1, "content_hash": "sha256:3a0ffd7c0e6eeb8dbb405824b2b4da7b562222037582ba6fa46aedd392ad1a7b" }, "sections": [ { "title": "A pattern can hide inside a coloured word", "text": "Take an alphabet with two letters, A and B. A word is just a finite string such as ABA. Colour each possible word red or blue. A combinatorial line keeps some positions fixed and changes a nonempty shared set of positions together. For example, A?A gives AAA and ABA when the question mark is replaced by A and then B." }, { "title": "The theorem is about every colouring", "text": "Hales and Jewett show that any fixed finite alphabet and finite colour count have a sufficiently large word length at which every colouring contains a one-colour combinatorial line. The variable positions all receive the same letter. The length can depend on the alphabet and number of colours. This statement is stronger than observing a pattern in one pleasant example." }, { "title": "Try the smallest two-colour case", "text": "With length one, colour A red and B blue: the only line is not one colour. With length two, consider AA, AB and BB. Each pair forms a line: change the second position, the first position, or both positions together. Three objects coloured with only two colours must have a same-colour pair. Therefore every two-colouring of all four length-two words already contains a line. There are sixteen possible two-colour assignments to those four words; a short program can check them all." }, { "title": "Why the game connection needs care", "text": "A sufficiently large board with no draw colouring can imply a first-player strategy in the positional game studied by the paper. That does not say every familiar small board lacks draws. Nor does a brute-force check of the binary length-two example prove the general theorem for every alphabet and colour count. A result can be powerful because its quantifiers cover all those finite choices, even when particular tiny cases are easy." }, { "title": "Leave a check someone can reuse", "text": "Write a program that generates legal lines, then enumerates the sixteen tiny colourings. Save the line-generation convention and a failing test for an invalid line whose variable positions change independently. Report the exact finite case checked. The useful contribution is a transparent teaching or verification artifact, together with its limits; it is not a new proof of the unrestricted Hales–Jewett theorem." } ], "references": [ { "title": "Regularity and Positional Games", "url": "https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143712-1/S0002-9947-1963-0143712-1.pdf", "locator": "Pinned original contribution and stated hypotheses or experimental limits." } ] }, "reading_status": "EXPLANATORY_GUIDE" }