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Claim. The note Weighted sunflower pressure and the exact polylogarithmic exponent in harmonic LCM patterns (Chojecki, dated 15 April 2026, posted the same day in the discussion thread of Problem 856 with the statement that it was developed with GPT-5.4 Pro) claims an exact exponent for $f_k(N)$, the largest harmonic sum of a set $A\subseteq\{1,\ldots,N\}$ with no $k$ distinct elements whose pairwise least common multiples all agree. For $z>0$ it defines the weighted partition function

$$W_k(n;z)=\max\Bigl\{\sum_{S\in\mathcal F}z^{|S|}:\ \mathcal F\subseteq2^{[n]} \text{ has no }k\text{-sunflower}\Bigr\}, \qquad \Lambda_k(z)=\lim_{n\to\infty}W_k(n;z)^{1/n},$$

proves the two weighted bounds $f_k(N)\le(\log N)^{\Lambda_k(z)-z+o(1)}$ and $f_k(N)\ge(\log N)^{\log(z\Lambda_k(1/z))/z-o(1)}$ for every fixed $z>0$, and concludes by a squeezing argument that

$$f_k(N)=(\log N)^{\gamma_k+o(1)},\qquad \gamma_k=\lim_{z\to\infty}\bigl(\Lambda_k(z)-z\bigr) =\inf_{z>0}\bigl(\Lambda_k(z)-z\bigr).$$

At $z=1$ the bounds reduce to those of Tang and Zhang (claim page; $\Lambda_k(1)$ is their sunflower-free capacity $\mu_k^S$), and the note says that $\gamma_k=1$ when $\mu_k^S=2$. The upper bound transports the harmonic weight along squarefree multipliers chosen with probability proportional to $z^{\omega(q)}/q$, so that the multiplier histories landing at one endpoint form a sunflower-free family; the lower bound is a weighted bucketing construction. The note states that computing $\gamma_k$, or expressing it through $\mu_k^S$, remains open.

Submission note. Posted to the site's forum by Przemek Chojecki on 15 April 2026:

With GPT-5.4 Pro I've developped a weighted version of the Tang-Zhang sunflower-capacity argument that gives the exact estimate. The written note is here.

The starting point is the same mass-transport idea that Liam Price emphasized in his recent Markov-chain discussion of [1196]: after truncating to squarefree multipliers $q\le N$, one chooses $q$ with probability proportional to $z^{\omega(q)}/q$, so that the harmonic weight $1/a$ is transported to the endpoint $m=aq$ with density proportional to $1/m$. The admissible multiplier histories landing at a fixed endpoint form a $k$-sunflower-free family, and this yields a weighted upper bound involving the partition function $W_k(n;z)$.

Covers. If it stands, $f_k(N)=(\log N)^{\gamma_k+o(1)}$ with $\gamma_k=\inf_{z>0}(\Lambda_k(z)-z)$: the existence of an exact exponent and its characterization through the weighted sunflower-free partition function. 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 note leaves that value open on its own account, so the claim is recorded as partial although the note presents the exponent as the answer. The later claim of [[problems/integer_sequences/E0856/claims/2026_07_18_rayyoung_zhu_luo|RayYoung, Zhu and Luo]] asserts an exponent of the same shape through a different extremal quantity; the two are recorded separately, and no comparison of the two exponents is recorded.

Read depth. The claim is recorded from the note's abstract and introduction (pp. 1--2); its proofs are not assessed. Nothing here is this project's own review.

Standing. Claimed: a note on the author's site with no arXiv or journal record found. In the thread on 15 April 2026 the site's curator restated the claim's main statement after unpacking the notation, said that the details had not been checked, and recommended a rewrite; another contributor reported that an automated check found no issues and no prior literature, which is a thread comment, not a review of record; a comment of 1 July 2026 reports an attempt, with the AI system Aristotle, to formalize the argument in Lean that stalled on a missing uniform sunflower lemma and the lower capacity bound. The site's label is OPEN (as of 2026-10-07; page last edited 18 January 2026), and the claim was not entered on the proof-claim tab.

Depends on. No page of this wiki.