Erdős Problems: discussion thread for Problem 963

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Erdős Problems contributors, "Thread for problem 963," community discussion, posts through 3 September 2026, captured 6 September 2026. https://www.erdosproblems.com/forum/thread/963

Read status. Claims checked. The complete thread was read, including the proposed proofs, objections, corrections, and later finite computations. No proof in the thread was verified by this reading. The linked ChatGPT shares are external to the retained artifact and were not assessed.

Claimed asymptotic bound

In post 2027, KoishiChan claims that every $n$-element set of reals contains a dissociated subset of size $(1-o(1))\log_2 n$. After asserting reductions first to integers and then to positive integers, the post defines the positive-integer extremal function and proposes a recursion of the shape

$$f(n)\geq \frac{1}{12}\log_2 n+ f\!\left(\frac{n^{11/12}}{\log^4 n}\right).$$

The proposed proof chooses $p\asymp n^{1/12}$, dilates the set modulo a larger prime $q$, and uses a character-sum second-moment estimate to find well-populated progression cells in every residue class modulo $p$. It then combines a dissociated residue set with a dissociated subset in the zero class and iterates the resulting recursion. This is an author claim in a discussion post, not an established theorem in this corpus.

The replies record the following review history:

• post 2030 suggests that primality of $p$ may be unnecessary and that choosing near a power of two could improve constants; it does not verify the proof;
• post 2033 identifies an off-by-one defect in the claim that every selected lift lies in $[1,q/k]$;
• in post 2046, KoishiChan says the defect is fixed by lowering the upper bound on the progression parameter by one, and post 2047 accepts that response. The corrected definitions and all downstream estimates are not written out in the thread;
• post 2049 reports a ChatGPT Pro review that found minor issues and possibly the same semi-major issue, and recommends a maintained formal write-up; and
• post 3664 says the argument looks good, proposes marking the catalog problem solved, and asks for a formal PDF. This is a favorable informal assessment, not a published or independently verified proof. The unanswered question in post 6885 leaves the thread unclear about whether a defect or the missing write-up prevented a later status update.

Elementary sign-class step

The positive-integer reduction in post 2027 is not explained there. One part of it has a short self-contained justification. For an $n$-element integer set, discard $0$ if present and take the larger of its positive and negative classes; that class contains at least $\lceil(n-1)/2\rceil$ elements. Negating the whole negative class preserves every equality or inequality between subset sums. Thus a lower bound valid for all $m$-element sets of positive integers gives the same bound for an integer subset of size $m=\lceil(n-1)/2\rceil$. This constant-factor loss is harmless for a claimed $(1-o(1))\log_2 n$ estimate. This argument only supplies the integer-to-positive sign-class step; the preceding real-to-integer reduction is asserted, but not proved, in post 2027.

Rejected exact proof

Post 3658 claims the exact bound $f(n)\geq\lfloor\log_2 n\rfloor$. Its essential unsupported lemma is that the initial interval $\{1,\ldots,n\}$ minimizes the largest dissociated subset among all $n$-element real sets. The powers-of-two construction proves a lower bound only inside that particular interval; it gives no implication for an arbitrary set. Post 3662 rejects the interval-extremality assertion as unjustified and links a separate AI review. The later positive-integer example in posts 8701 and 8709 disproves that lemma. It does not disprove the exact logarithmic conjecture, so the failed argument must not be reported as either a proof or a disproof of Problem 963.

Corrected 13-element example

In post 8701, BAKKAOUI gives

$$A^*=\{1,2,3,4,5,6,7,8,9,10,12,13,15\}$$

and reports $d(A^*)=4$ while $d(\{1,\ldots,13\})=5$. The post describes an exact-integer enumeration of all $\binom{13}{5}=1287$ five-subsets of $A^*$, none dissociated, and gives $\{1,2,4,8\}$ as a dissociated four-subset. Hence the reported fixed example yields $f(13)\leq4$ and refutes universal extremality of the initial interval. Since $\lfloor\log_2 13\rfloor=3$, it does not refute the conjectured lower bound.

Post 8709 supplies three necessary corrections and qualifications: heredity of dissociation turns the exhaustive five-subset check into an upper bound for all larger subsets; the search over $13$-element subsets of $\{1,\ldots,34\}$ proves only a bounded window minimum, not $f(13)=4$ over all real sets; and the claimed smallest counterexample applies only to positive integers, because adjoining $0$ already defeats interval extremality at $n=4$. The larger window searches through $n=16$, the proposed exceptional role of $n=13$, and the negative OEIS and literature reports remain source-reported computations, not general theorems.

The fixed comparison has a separate [[additive_combinatorics/bakkaoui_2026_dissociated_interval_counterexample/_index|source record and bounded reconstruction]]; that record, rather than the forum's broader search report, states the local evidence and review standing for the finite instance.

AI and computation provenance

• Post 2049 discloses an AI review of KoishiChan's argument; its linked review is not contained in this capture.
• Post 3658 labels its exact proof as AI-assisted. In post 3671, the named model is questioned and tentatively corrected. Post 3662 also links an external AI review.
• Post 8701 says AI agents assisted both the finite searches and the literature check, while BAKKAOUI personally checked the displayed counterexample in exact integer arithmetic. The thread includes no executable search artifact, so its wider computational claims retain their stated bounded scope.
• Post 1593 reports a Google Scholar and ChatGPT literature search and calls the problem open. That search report alone does not establish current mathematical status.

Bears on. Problem 963. The asymptotic and exact-resolution arguments remain community claims with the qualifications above; the corrected finite example bears only on interval extremality.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.