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Claim. A manuscript by RayYoung, Keheng Zhu and Yanping Luo, written with the AI system GPT 5.6 Sol Pro and entered on the proof-claim tab of Problem 856 on 18 July 2026 as a full claim, states that

$$f_k(N)=(\log N)^{\gamma_k+o(1)},\qquad \gamma_k=\sup_{n\ge1,\ 1\le r\le n}\frac{r}{en}\,M_k(n,r)^{1/r},$$

where $M_k(n,r)$ is the largest size of a family of $r$-element subsets of $\{1,\ldots,n\}$ containing no $k$ sets with the same pairwise union. The tab's summary describes the argument as a pair of weighted bounds, upper and lower, that match in the exponent, followed by a reduction of the variational quantity they produce to this extremal quantity on finite ground sets. The authors' note on the tab says that they checked the argument themselves, that the proof is short and involves little computation, that no Lean formalization accompanies it, that the work builds on the literature and on the ideas discussed in the problem's thread, and that refining $\gamma_k$, including for particular $k$, is tied to the sunflower conjecture of Problem 857 and left open.

Submission note. Posted to erdosproblems.com as a proof claim by RayYoung, Keheng Zhu, Yanping Luo (account RayYoung) on 18 July 2026, giving "GPT 5.6 Sol Pro" as the AI used:

We prove that

$$> f_k(N)=(\log N)^{\gamma_k+o(1)}, \qquad \gamma_k= \sup_{\substack{1\le r\le n}} \frac{r}{en}M_k(n,r)^{1/r}, >$$

where $M_k(n,r)$ is the largest size of an $r$-uniform family on $[n]$ containing no $k$ sets with the same pairwise union. The proof obtains matching weighted upper and lower bounds and then reduces the resulting pressure formula to this finite-block extremal formula. Notes: After obtaining this result, we checked the argument ourselves. Since the proof is relatively straightforward and involves little computation, we uploaded the manuscript after only minor revisions, without providing a corresponding Lean formalization. The paper builds on existing literature and the ideas discussed in this thread; we regard it as a natural continuation of these contributions and are grateful to everyone who has worked on the problem. We also attempted to refine the estimates for $\gamma_k$, including for specific values of $k$, but this appears to be closely related to Problem #857 and the sunflower conjecture, and seems to require further investigation.

Covers. If it stands, $f_k(N)=(\log N)^{\gamma_k+o(1)}$ with $\gamma_k$ the supremum displayed above: the existence of an exact exponent and its characterization through the extremal numbers $M_k(n,r)$. Not covered: the value of $\gamma_k$. The problem asks for an estimate of $f_k(N)$, which for a function of polylogarithmic growth is its exponent, and the authors leave that value open, tying it to the sunflower conjecture, so the claim is recorded as partial although the tab enters it as a full claim. The exponent has the same shape as the one in [[problems/integer_sequences/E0856/claims/2026_04_15_chojecki|Chojecki's earlier claim]], through a different extremal quantity; the two claims are recorded separately, and no comparison of the two exponents is recorded. Both refine the bounds $(\log N)^{\log\mu_k^S-o(1)}\le f_k(N)\ll(\log N)^{\mu_k^S-1+o(1)}$ of Tang and Zhang (claim page).

Read depth. The claim is recorded from the proof-claim tab's summary and the authors' note; the manuscript at the Overleaf link is not assessed. Nothing here is this project's own review.

Standing. Claimed: a shared Overleaf manuscript with no arXiv or journal record found. As of 2026-10-07 the proof-claim tab shows the claim with no comments and the site's standing notice that appearing on the tab means no one at the site has examined the proof; the site's label is OPEN (page last edited 18 January 2026).

Depends on. No page of this wiki.