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Claim. Call a family of subsets of $\{1,\ldots,n\}$ admissible for $r$ when no member contains another and every size that occurs among its members occurs at least $r$ times, let $g(n,r)$ be the largest number of distinct sizes an admissible family can have, and let $n_0(r)$ be the least $N$ such that $g(n,r)=n-3$ for every $n>N$. Appendix A of the paper computes

$$n_0(2)=3,\qquad n_0(3)=8,$$

and for larger $r$ the paper bounds the threshold: Theorem 1.4 gives $n_0(r)\ge 2r+2$ for every $r\ge4$, and Theorem 1.5 gives $n_0(r)\le 2r+2\log_2 r+O(\log_2\log_2 r)$ for every $r\ge2$, so that $n_0(r)=2r+o(r)$. Under the reading in the Formulation of Problem 776, which asks for estimates of $n_0(r)$, these results answer the question; they determine $n_0(r)$ exactly only at $r=2,3$, and the exact values for $r\ge4$ are the subject of [[problems/set_systems/E0776/claims/2026_07_17_thiim|Thiim's determination of the threshold]] and Ronen's value n_0(4)=12, which rest on this page. Remark 1.2 notes that requiring exactly $r$ sets of each occurring size, instead of at least $r$, gives the same $g(n,r)$. The tools are estimates on central binomial coefficients (Lemma 2.1 and Corollary 2.2, the latter stating that the least $m$ with $\binom{m}{\lfloor m/2\rfloor}\ge K\ge4$ satisfies $m\le\lceil\log_2 K+\tfrac12\log_2\log_2 K+2\rceil$) and explicit constructions that fill the levels of the Boolean lattice; the constructions and the exhaustive search of Appendix A are posted in the companion repository linked above. The source card holds the digest.

Submission note. Posted to the site's forum by Quanyu Tang on 11 February 2026:

1. This problem also appears in the following volume, which seems to be the original source: P. Erdős, Problem sessions, In: Ordered Sets (Proc. NATO Adv. Study), edited by I. Rival, Dordrecht: Reidel (1981), 860--861.

2. In [Gu83], Guy writes: "[$\ldots$] We have no satisfactory estimate of $n_0(r)$. The content of the previous two paragraphs will be a small subset of a forthcoming paper of Erdős, Szemerédi and Trotter." However, I have not been able to locate any related publication in MathSciNet, Google Scholar, the Erdős paper website, or on Trotter's personal homepage.

3. Let $n_0(r)$ be the threshold such that whenever $n>n_0(r)$ one can achieve $n-3$ distinct set sizes in such a family. My friend He and I have just written a paper (arXiv:2602.09803v1). By iterating ChatGPT-5.2 Thinking hundreds of times, we made some progress on this problem: $n_0(2)=3$ and $n_0(3)=8$, and moreover for every integer $r\ge 4$ one has

$$2r+2 \ \le\ > n_0(r)\ \le\ 2r+2\log_2 r + O(\log_2\log_2 r).$$

(The site has been updated to address this comment.)

Claimant. Yixin He and Quanyu Tang, An Erdős–Trotter problem on antichains with multiplicity $r$ on each occurring level, arXiv:2602.09803, first version 10 February 2026, second version 21 March 2026, which the arXiv comment marks as the submitted version; the record lists no journal reference. Tang announced the paper in the problem's forum thread on 11 February 2026 and wrote that the results were obtained by iterating ChatGPT-5.2 Thinking; the site's commentary credits the paper with ChatGPT.

Acceptance. None that counts as evidence. The site labels the problem OPEN (page last edited 10 April 2026), although its commentary credits He and Tang [HeTa26b] with the two values and the bounds. The site's curator, Thomas Bloom, wrote in the thread on 10 April 2026 that, the problem being loosely phrased, Bloom was minded to mark it solved and asked for the views of the authors and others; the label was not changed. Commentary on a problem the site labels OPEN is not acceptance, so no reviewed evidence is listed; the paper has no journal record, so no refereed evidence; and no Lean audited by the corpus checks the computation, so no formalized evidence.