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Claim. In the notation of Problem 1183, Theorem 1.2 of the manuscript states

$$\left\lceil\frac{n+1}2\right\rceil\le F(n)\le n^{\log_2n+O(\log\log n)},$$

so that $F(n)=\exp(O((\log n)^2))=(1+o(1))^n$, and Theorem 1.3 states $\lceil(n+1)/2\rceil\le f(n)\le(3+o(1))n\log_2n$. The upper bound on $F$ identifies the free rank of a union-closed family, the largest number of members with all nonempty unions distinct, with the VC-dimension of a complementary set system, and rules out large free rank under a random coloring, the step Erdős's 1978 Theorem 2 takes for generators with distinct unions. The bound on $f$ represents a family closed under unions and intersections as the lattice of ideals of a poset (Birkhoff's theorem), rules out large Boolean cubes under a random coloring, and counts posets of bounded width. The manuscript's Questions 5.2 and 5.3 leave the orders of $f(n)$ and $F(n)$ open, including whether $F(n)$ is polynomial or quasipolynomial.

Covers. The second "in particular" question, whether $F(n)<(1+o(1))^n$: the answer is yes if Theorem 1.2 holds. The first question, whether $F(n)$ is superpolynomial, and the estimates of $f(n)$ and $F(n)$ are not settled by the claim; its upper bound on $f(n)$ is a new bound below the trivial $2^n$.

Depends on. Nothing in this wiki; the manuscript's arguments are self-contained apart from Birkhoff's representation theorem and the Sauer--Shelah bound.

Standing. Claimed. This page rests on the manuscript at the linked URL (abstract, Definition 1.1, Theorems 1.2 and 1.3, Proposition 2.1 and Section 5; 94,159 bytes); no proof was checked, so no evidence kind is listed. The author's thread comment of 18 March 2026 says the partial results were obtained with GPT-5.4 Pro; the manuscript carries no such declaration; that is provenance only. Later comments in the thread report comparisons of the two arguments made with GPT-5.4 Thinking and with Gemini 3.1 Pro, and a run of a checking tool the comments call Standard check, which found the argument plausible but not complete, with a few issues judged likely minor; those are not reviews. In the same thread a second group of three authors reports almost the same partial result, obtained with GPT-5.4 Pro about a day earlier, and posts its GPT-5.4 Pro chat history in a GitHub repository, a PDF of the chat record and an English translation of it; no manuscript was posted, chat records are not citable sources, and a thread post without a dated manuscript gets no claim page, so that report has none. No arXiv version, journal record or independent review was found on 2026-09-18; the site's label is OPEN and its commentary does not mention the manuscript, which is not filed in the library. Its reference for the 1978 source names a different Erdős paper.