---

Basile Beyer de Ryke, Arithmetic oscillations in a prescribed subset-sum problem of Erdős and Graham, a dated manuscript (a revised version dated 25 July 2026), posted on that day as a comment under Principia Math's proof claim (whose entry names GPT 5.6 and Opus 4.8 as the systems Principia Math used) on the tab of Problem 361. With $F_c(n)$ the largest size of $A\subseteq\{1,\ldots,\lfloor cn\rfloor\}$ with $n$ not a sum of distinct elements of $A$, and irregularity read as failure of $F_c(n)/n$ to converge, Theorem 1.1 states that for every fixed $0<c<1$ the sequence $F_c(n)/n$ does not converge, and more precisely, with $H(x)=\lfloor3/(2x)\rfloor$, that for $0<c\le1/2$

$$\limsup_{t\to\infty}\frac{F_c(2t)}{2t}\le\frac{1}{2H(c)}<\frac c2 \le\liminf_{t\to\infty}\frac{F_c(2t+1)}{2t+1},$$

with a corresponding statement for $1/2<c<1$. The abstract and the thread comment state the further results: an exact formula $F_c(n)=\lfloor cn\rfloor-\lceil n/2\rceil$ for $c\ge1$; upper bounds along arithmetic subsequences that depend on the small divisors of $n$, through the least positive integer not dividing a chosen divisor of $n$, attained asymptotically in several cases by the multiples of the least non-divisor, for instance $F_c(n)=n/K+o(n)$ along odd $n$ for $c=2/K$ (Proposition 5.3(1)); at $c=3/4$, the limits $F_{3/4}(n)/n\to3/8$ along odd $n$, with the upper bound from a pairing argument (Proposition 5.1), and $\to1/3$ along even $n$ not divisible by $3$, with the upper bound from a reflection inequality and attained by the multiples of $3$ together with one residue class modulo $3$ above $n/2$, a construction the note credits to a thread comment of 17 October 2025 (Proposition 5.3(2)); arbitrarily many distinct subsequential densities in the original fixed-parameter formulation; and the two parity limits at $c=1/2$ (Proposition 5.2), of which the even one, $1/6$, is Alon's Corollary 2.6 of 1987, as the note says (claim page), and the odd one, $1/4$, is the note's own. The tools are Alon's short zero-sum theorem with extraction and pairing arguments. The note says that the exact behavior for general $c$ and $n$ remains open.

Covers. The second question, answered yes for every $0<c<1$, with explicit subsequential limits in several arithmetic classes, and the first question exactly for $c\ge1$; it does not determine $F_c(n)$ for general $c$ and $n$. It overlaps Principia Math's claim of two days earlier, which the thread compares with it; the two are independent write-ups.

Read depth. The account above rests on the note's statements; its proofs are not checked on this page, and nothing here is this project's own review.

Standing. Claimed: a note on a file-sharing service whose identity is not pinned, posted as a thread comment and not as a proof claim of its own; the site's label is OPEN (page last edited 17 October 2025; thread accessed 2026-10-07), and no preprint-server version, refereed version, site acceptance or independent review was found on 2026-10-07.

Depends on. Alon 1987, Corollary 2.6, for the even parity limit at $c=1/2$.