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Claim. Quanyu Tang and Shengtong Zhang, Harmonic LCM patterns and sunflower-free capacity (arXiv:2512.20055, version 1 of 23 December 2025; library card tang_2025_harmonic_lcm_patterns_sunflower_free_capacity), prove bounds for the function $f_k(N)$ of Problem 856, for each fixed $k\ge3$:

• Theorem 1.2: $f_k(N)\ge(\log N)^{c_k-o(1)}$ with $c_k=(k-2)/\bigl(e((k-2)!)^{1/(k-2)}\bigr)$, by a construction that splits the primes into blocks of comparable harmonic sum and takes squarefree integers with exactly $k-2$ prime factors from each block.
• Theorems 1.5 and 1.6: with $\mu_k^S=\lim_n F_k(n)^{1/n}$ the sunflower-free capacity, where $F_k(n)$ is the largest family of subsets of an $n$-element set with no $k$-sunflower, $(\log N)^{\log\mu_k^S-o(1)}\le f_k(N)\ll(\log N)^{\mu_k^S-1+o(1)}$.
• Theorem 1.4: $\mu_k^S=2$, that is, the sunflower conjecture of Problem 857 fails at $k$, if and only if $f_k(N)=(\log N)^{1-o(1)}$.
• Corollary 1.7: $(\log N)^{\log1.551-o(1)}\le f_3(N)\ll(\log N)^{3/2^{2/3}-1+o(1)}$, from known bounds for $\mu_3^S$.

The $k=3$ case of Theorem 1.2, $f_3(N)\gg(\log N)^{1/e-o(1)}$, appeared first as Theorem 2.1 of Tang's note A note on Erdős Problem #856, dated 10 December 2025 and linked from Tang's thread posts of 9 and 10 December 2025; the note's repository describes it as a preliminary write-up superseded by the paper.

Submission note. Posted to the site's forum by Quanyu Tang on 24 December 2025:

In joint work with Shengtong, we wrote a paper on this problem (arXiv:2512.20055) and make precise its connection to [857]. Summary: Define the Erdős-Szemerédi $k$-sunflower-free capacity by (\mu_k^{\mathrm S}:=\limsup_{n\to\infty} F_k(n)^{1/n}), where $F_k(n)$ denotes the maximum size of a $k$-sunflower-free family of subsets of $[n]$. [857] (the Erdős-Szemerédi sunflower conjecture) asserts that $\mu_k^{\mathrm S}<2$ for every $k\ge 3$. Our main results are: (1) Equivalence. For each fixed $k\ge 3$, $ \mu_k^{\mathrm S}=2$ iff $ f_k(N)=(\log N)^{1-o(1)}.$

(2) Lower bounds. $f_k(N)\ge (\log N)^{b_k-o(1)}$, where $b_k:=\max{c_k, \log\mu_k^{\mathrm S}}$ and $c_k:=\frac{k-2}{e((k-2)!)^{1/(k-2)}}.$

(3) Upper bound. $f_k(N)\ll (\log N)^{\mu_k^{\mathrm S}-1+o(1)}.$

As an illustration, when $k=3$ one has $\mu_3^{\mathrm S}\ge 1.551$ (a construction of Deuber--Erdős--Gunderson--Kostochka--Meyer) and $\mu_3^{\mathrm S}\le 3/2^{2/3}$ (Naslund--Sawin), hence

$$(\log > N)^{\log(1.551)-o(1)} \ \le\ f_3(N)\ \ll\ (\log N)^{\frac{3}{2^{2/3}}-1+o(1)}. >$$

Numerically, $\log(1.551)\approx 0.4389$ and $\frac{3}{2^{2/3}}-1\approx

0.8899$.

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

Covers. The lower and upper bounds above and the equivalence of Theorem 1.4. Not covered: the value of the exponent of $f_k(N)$, which the bounds leave open for every $k$.

Standing. Claimed: the paper has an arXiv record only. The site's commentary credits the bounds to Tang and Zhang on a problem it labels OPEN; that credit is commentary, not acceptance.

Depends on. No page of this wiki.