Every Seven Positive Reals Contain a Dissociated Four-Element Subset: Small Cases of an Erdős Problem on Dissociated Sets
Year: 2026
Type: preprint
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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: EP-963 (Erdős Problem #963; P. Erdős, Extremal problems in number theory, Proc. Sympos. Pure Math. 8 (1965), p. 188).
Author: Ferudun, Alper
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222301007
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