{
  "schema_version": 1,
  "problem_number": "EP-963",
  "title": "Every Seven Positive Reals Contain a Dissociated Four-Element Subset: Small Cases of an Erdős Problem on Dissociated Sets",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let f(n) be the largest k such that every set of n real numbers contains a dissociated subset of size k, that is, a subset whose 2^k subset sums are pairwise distinct (the empty sum included). Erdős noted that f(n) ≥ ⌊log₃ n⌋ and asked whether f(n) ≥ ⌊log₂ n⌋ for every n (Erdős Problem #963). The general question remains open. Let g⁺(K) be the largest size of a set of positive reals without a dissociated subset of size K. We show that the corresponding largest size for sets of reals is 1 + 2g⁺(K), so that the question is equivalent to g⁺(K) ≤ 2^(K−1) − 1 for all K, and also to the inequality |P| ≤ 2^(dim P) − 1 for finite sets P of nonzero integer vectors containing no pair x, −x, where dim P is the largest size of a dissociated subset of P. Elementary arguments give g⁺(1), g⁺(2), g⁺(3) = 0, 1, 3. Our main result is g⁺(4) = 6: every seven positive reals contain a dissociated four-element subset, and {1, …, 6} contains none. The proof is computer-assisted and is given twice: by a case analysis over sign patterns in which every terminal case is closed by an exact Farkas certificate, checked by independent exact checkers, and by an enumeration of linear forms over a dissociated triple in exact integer arithmetic. Consequently f(n) is known for all n ≤ 27: it is 0, 1, 1, 2, 2, 2, 2 for n ≤ 7, then 3 for 8 ≤ n ≤ 13 and 4 for 14 ≤ n ≤ 27; and f(n) ≥ ⌊log₂ n⌋ holds for all n ≤ 31. The first open case, n = 32, is equivalent to g⁺(5) ≤ 15. The upper bound for 14 ≤ n ≤ 27 uses the set {1, …, 10, 12, 13, 15} without dissociated five-element subsets, posted by BAKKAOUI on the problem's forum (2026); we also show g⁺(6) ≥ 24 by an explicit set. An unrefereed argument for f(n) ≥ (1 − o(1)) log₂ n was posted on the same forum by KoishiChan (2025). To our knowledge the value g⁺(4) = 6 and its consequences for n ≤ 31 are new. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "keywords": [
    "dissociated sets",
    "distinct subset sums",
    "additive dimension",
    "Erdős problems",
    "Erdős Problem 963",
    "exact small values",
    "computer-assisted proof",
    "Farkas certificates",
    "partial answer",
    "UnsolvedMath",
    "EP-963",
    "math.CO",
    "math.NT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/ep-963/",
  "pdf_url": "https://eulersolve.org/papers/ep-963/paper.pdf?v=299a667f3994",
  "doi": "10.5281/zenodo.23396928",
  "zenodo_record_url": "https://zenodo.org/records/23396928",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "On Erdos Problem 963 (dissociated subsets of sets of n real numbers): every seven positive reals contain a dissociated four-element subset, proved twice by exact computer-assisted methods; hence f(n) is known exactly for n at most 27 and f(n) is at least floor(log2 n) for every n at most 31. The general question remains open; the first open case is n = 32.",
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    "source.zip": {
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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."
}