Open. Writing $f_c(n)$ for the largest size, Alon's refereed Corollary 2.6 of 1987 gives $f_{1/2}(n)=(1/6+o(1))n$ along even $n$ (claim page (Alon, 1986)), and three partial claims of 2026 are pending: a proof claim filed as full by Principia Math (using GPT 5.6 and Opus 4.8, as the proof-claims tab names them) on 23 July 2026, whose theorem says that $f_c(n)/n$ does not converge for $0<c<1$ and that $f_c(n)=\lfloor cn\rfloor-\lceil n/2\rceil$ for $c\ge1$, and which its thread records, with the claimant's agreement, as answering the second question and not the first (claim page (Principia Math, 2026)); Beyer de Ryke's note of 25 July 2026 in the same thread, proving the same non-convergence with explicit limits along arithmetic subsequences (claim page (Beyer de Ryke, 2026)); and Principia Math's second result, announced in the thread on 8 August 2026, an inverse zero-sum theorem giving $f_c(n)=(c/\operatorname{snd}(s)+o(1))n$ along the multiples $n$ of $s$ not divisible by the least non-divisor $\operatorname{snd}(s)$ of $s$, for $0<c\le1/s$, with its representation theorem for $s=2$ and $1/3<c<1/2$ stated in Lean (claim page (Principia Math, 2026)). The site's label is OPEN (page last edited 17 October 2025; thread and proof-claims tab accessed 2026-10-07).