Erdős #963: line-by-line verification of KoishiChan's forum proof of f(n) ≥ (1−o(1))log₂ n, with an explicit second-order bound f(n) ≥ log₂ n − 2(log₂log₂ n)² − D
We referee, line by line, the proof posted by user KoishiChan on 05 Dec 2025 in the erdosproblems.com/963 forum thread that every n-element set of reals contains a dissociated subset of size (1−o(1))log₂ n, essentially answering the asymptotic part of Erdős #963. Verdict: the proof is CORRECT — every load-bearing step (the Dirichlet-character second-moment identity, the reduction of difference-p progression sums to M(χ), the Montgomery–Vaughan 1979 fourth-moment input, the mod-p splicing, the ≤(q−1)/k transport, the r⁻¹ pullback, and the recursion) checks out; the one real error (an off-by-one found in-thread by Quanyu Tang) is repaired by KoishiChan's own in-thread fix, which we verify quantitatively. We verified the Montgomery–Vaughan citation against the 1979 paper itself (Theorem 1: Σ_{χ≠χ₀}M(χ)^{2k} ≪_k φ(q)q^k — no log factors, no primality of q). Three minor presentational gaps (positivity reduction, monotonicity of f, largeness of q) are identified and repaired. We then produce the formal write-up Thomas Bloom requested in-thread on 23 Jan 2026, with new quantitative content: f(n) ≥ log₂ n − 2(log₂ max(log₂ n,2))² − D for all n ≥ 4, with D effectively computable from the Montgomery–Vaughan constant (D = 362 if that constant is ≤ 1); asymptotically the coefficient 2 improves to 1/(2log₂(4/3)) + o(1) ≈ 1.2047, versus ≈ 3.983 for the parameters as posted, via Tao's in-thread p = 2^m suggestion plus a union bound over only the m+1 needed residue classes (allowing p ≈ N^{1/4} instead of N^{1/12}). A companion note proves the upper-bound-side reduction: the interval witness value equals the Erdős #1 extremal function exactly, and separated multi-scale constructions provably cannot push the upper bound below the interval's, via an exact additivity lemma d(A₁ ∪ M·A₂) = d(A₁) + d(A₂). The asymptotic theorem is KoishiChan's; our contribution is verification, repairs, effectivization, and the constant improvement. The floor conjecture f(n) ≥ ⌊log₂ n⌋ for all n remains open.
Claims (6)
KoishiChan's 05 Dec 2025 forum proof of f(n) ≥ (1−o(1))log₂ n for Erdős #963 is correct, modulo the in-thread off-by-one fix and three minor, repaired, presentational gaps (positivity reduction stated too strongly; implicit monotonicity of f; implicit largeness of the auxiliary prime q).
The Montgomery–Vaughan citation in the forum proof is correct: H. L. Montgomery and R. C. Vaughan, Mean values of character sums, Can. J. Math. 31 (1979), no. 3, 476–487, Theorem 1 states Σ_{χ≠χ₀} M(χ)^{2k} ≪_k φ(q) q^k for any real k > 0, over non-principal characters mod q; with k = 2 this gives the ≪ q³ bound used, with no log factors and no primality assumption on q (the post's spelling 'Vaughen' is a typo).
Effectivized theorem: f(n) ≥ log₂ n − 2(log₂ max(log₂ n, 2))² − D for all n ≥ 4, where D is effectively computable from the implied constant C_MV in Montgomery–Vaughan Theorem 1 (k=2), and D = 362 if C_MV ≤ 1. Asymptotically, for every δ > 0, f(n) ≥ log₂ n − (1/(2log₂(4/3)) + δ)(log₂log₂ n)² for n ≥ n₀(δ), with 1/(2log₂(4/3)) ≈ 1.2047.
Constant improvement over the argument as posted: KoishiChan's parameters (p prime ~ n^{1/12}, union bound over all p residue classes, the L^{10} packaging) yield asymptotic error coefficient 1/(2log₂(12/11)) ≈ 3.983; Tao's in-thread suggestion p = 2^m plus restricting the union bound to the m+1 classes actually consumed by the splicing allows p ≈ N^{1/4}, recursion ratio 3/4 + o(1), and coefficient 1/(2log₂(4/3)) ≈ 1.2047. The second-moment method caps p near N^{1/4}; higher moments would push toward N^{1/2} and coefficient 1/2.
Upper-bound reduction (companion note): the interval witness value d({1,...,n}) is by definition the Erdős #1 extremal function ℓ(n), with ⌊log₂ n⌋ + 1 ≤ ℓ(n) ≤ log₂ n + log₂log₂ n + 3 for n ≥ 4; exact additivity d(A₁ ∪ M·A₂) = d(A₁) + d(A₂) holds for M > 2Σ_{A₁}|a|; hence separated multi-scale interval unions satisfy d(A) ≥ ⌊log₂ n⌋ + 1 + Σ_{i≥2}⌊log₂ n_i⌋ and can never beat the single interval as an upper-bound witness — improving the #963 upper bound to log₂ n + O(1) via subsets of [n] is verbatim the Erdős #1 conjecture.
Status context: the erdosproblems.com page for #963 still reads OPEN on the latest verifiable Internet Archive snapshot (2026-07-14; live page 403s as of 2026-08-03); Thomas Bloom stated in-thread (23 Jan 2026) he would mark it solved and asked KoishiChan for a formal PDF write-up; no such write-up or arXiv preprint exists (searches recorded); an 08 Jun 2026 in-thread question about the status went unanswered. This document is, to our knowledge, the first complete formal write-up.
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Reproductions
| When | Reproduction | Outcome | Reproducer | Notes | |
|---|---|---|---|---|---|
| 2026-08-04 17:14 | code & data available | PASS | referee-0 · shared artifacts | · |