Wiki
Wiki

Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated


Claim. Write L(x)=exp⁡(log⁡xlog⁡log⁡x)L(x)=\exp(\sqrt{\log x\log\log x}) with natural logarithms, f(N)f(N) for the largest size of a family of pairwise disjoint residue classes with distinct positive moduli at most NN, and, for an integer m≥1m\ge1, ϵm\epsilon_m for the supremum of ∑i1/ni\sum_i1/n_i over finite disjoint families with distinct moduli m<n1<⋯<nkm<n_1<\cdots<n_k. Theorem 1 of Malek Zribi's note A Conditional Sharp Estimate for Erdős Problem 1190, dated 28 April 2026 and compiled as Zribi 2026, states that if f(N)=NL(N)−1+o(1)f(N)=NL(N)^{-1+o(1)} as N→∞N\to\infty, then

ϵm=L(m)−1+o(1)\epsilon_m=L(m)^{-1+o(1)}

as m→∞m\to\infty, which is the sharp estimate for the corrected Statement of Problem 1190. The note uses a supremum and remarks that the choice between a supremum and a maximum does not affect the asymptotic. Its argument is a reduction: the upper bound by Abel summation with A(x)≤f(x)A(x)\le f(x) for the counting function of a family's moduli, the lower bound from an extremal family for f(⌊mL(m)2⌋)f(\lfloor mL(m)^2\rfloor) with its at most mm small moduli discarded, and two lemmas on the integral of 1/(tL(t)α)1/(tL(t)^\alpha) and on the stability of LL under a change of scale. No structural information from a proof of the hypothesis is used. The note was posted on the site's discussion thread on 2026-04-28 by its author, who credits GPT-5.5 with the joint writing; the note names no journal, preprint server or identifier.

Hypothesis. The sharp asymptotic f(N)=NL(N)−1+o(1)f(N)=NL(N)^{-1+o(1)} for Problem 202, assumed and not proved in the note. The note itself gives no bound without it. The hypothesis is the theorem of Ho's manuscript, which also proves the unconditional transfer that this note expounds separately; so the note's conclusion is the estimate recorded on Ho's claim page, and this page records a second exposition of the transfer, not a second result.

Depends on. Ho's sharp asymptotic for Problem 202, the page where the hypothesis is established; the note depends on the statement, not on its proof.

Standing. Claimed. No outside reviewer has published an examination, the site records the problem's solution through Ho's manuscript and formalization rather than this note, and the site's curator has not credited it; the library card compiles the note's Theorem 1 and its argument without a verification record. The estimate it concludes concerns the supremum of the corrected Statement; the site's maximum is attained by no finite family.