{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0085",
  "title": "Essential Periods and Monochromatic Triangles in Measurable Layered Colorings",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let f: R -> {-1,1} be measurable and color the plane by chi(x,y)=f(y). Call an equilateral side length essentially avoided when, for each fixed orientation, the set of monochromatic translations has planar measure zero. We prove that essential avoidance of side s forces the essential period group of f to be exactly sqrt(3)s Z. Consequently at most one equilateral side length is essentially avoided. The classical EGMRSS side-length transfer then gives every nonequilateral three-point configuration a positive-measure set of monochromatic translations in some orientation. This includes collinear triples and survives arbitrary changes on a planar null set. The argument uses closed essential-period groups and one tilted equilateral triangle. It requires no interval or locally finite boundary assumption on the measurable profile. This is a complete result for essentially layered measurable colorings, related to AIM-COMBINATORICS-0085 in ulamai/UnsolvedMath v1.6.0; it does not solve the conjecture for arbitrary planar two-colorings. The standard stripe example, classical side-length transfer and earlier polygonal results are credited. Four-page English preprint by Alper Ferudun, Mercury Software GmbH. Version 1.0, 17 October 2026. AI-assisted, self-audited and unrefereed; no independent review, proof-assistant verification or absolute-priority certification is claimed. The accompanying source archive includes portable exact algebra and finite regression checks. CC BY 4.0.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "keywords": [
    "Euclidean Ramsey theory",
    "measurable coloring",
    "layered coloring",
    "essential period",
    "AIM-COMBINATORICS-0085"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0085/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0085/paper.pdf?v=9f5aed5d37e8",
  "doi": "10.5281/zenodo.23426780",
  "zenodo_record_url": "23426780",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The theorem covers all essentially layered measurable two-colorings, not arbitrary planar colorings. The unrestricted source conjecture remains open. Classical side-length transfer and earlier polygonal results are credited. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute-priority certification is claimed.",
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      "sha256": "9f5aed5d37e8360a590c3c4b928c03dab549d3696fbc52b5bd2dd294bc35633d"
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    "source.zip": {
      "sha256": "79aee43d29a6450443ae5170a6d6f30809c92ae98a44765871dd8b10c5f6a394"
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    "verification_report.md": {
      "sha256": "eaf2dd4b5a7c5115e043c3414ef6643be6cc2f56e25b3c218c9418f35ceb12e7"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}