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Updated Oct 8

Problem 856

claims/: The 4 claim pages of Problem 856, one per claimant's result; the problem's standing derives from them.

Statement. Let $k\geq 3$ and $f_k(N)$ be the maximum value of $\sum_{n\in A}\frac{1}{n}$, where $A$ ranges over all subsets of $\{1,\ldots,N\}$ which contain no subset of size $k$ with the same pairwise least common multiple.

Estimate $f_k(N)$ .

Status. Open. The site's label is OPEN (; page last edited 18 January 2026). Four pending partial claims are recorded. Two bound $f_k(N)$ without determining its order: Erdős's bound $f_k(N)\ll_k\log N/\log\log N$ of 1970 (claim page), and the bounds of Tang and Zhang of December 2025, $(\log N)^{c_k-o(1)}\le f_k(N)\ll(\log N)^{\mu_k^S-1+o(1)}$ with $\mu_k^S$ the sunflower-free capacity, together with their proof that $f_k(N)=(\log N)^{1-o(1)}$ exactly when the sunflower conjecture of Problem 857 fails at $k$ (claim page). Two later claims each assert $f_k(N)=(\log N)^{\gamma_k+o(1)}$ with an exponent defined by an extremal problem and not evaluated: a note of 15 April 2026 posted in the discussion thread, written with GPT-5.4 Pro, whose exponent is the infimum over $z>0$ of the growth rate of a weighted sunflower-free partition function minus $z$ (Chojecki's claim page); and a manuscript entered on the proof-claim tab on 18 July 2026 as a full claim, written with GPT 5.6 Sol Pro, whose exponent is the supremum of $(r/(en))M_k(n,r)^{1/r}$ over uniform families with no $k$ sets of equal pairwise union (the page of RayYoung, Zhu and Luo). These two are recorded as partial claims: the question asks for an estimate of $f_k(N)$ , which for a function of polylogarithmic growth is its exponent, and each claim characterizes the exponent without evaluating it, its value left open on the claimants' own account (the note says that computing $\gamma_k$ remains open; the manuscript's authors tie it to the sunflower conjecture); what each covers is stated on its page. The site's curator restated the first of them in the thread without checking it; the second has no comment on the tab. None of the four claims has a journal record, and nothing is reviewed here. The standing in the frontmatter is open, derived from the pending partial claims, no full claim being recorded.

Source. erdosproblems.com/856, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #856, https://www.erdosproblems.com/856.

References.

[Er70] Erdős, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133.
[TaZh25b] Q. Tang and S. Zhang, Harmonic LCM patterns and sunflower-free capacity. arXiv:2512.20055 (2025).

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (15)

Harmonic sums of LCM-k-free sets are o(log N)Polylogarithmic bounds tied to the sunflower-free capacity Chojecki's exact exponent through weighted sunflower pressure RayYoung, Zhu and Luo's exact exponent by a finite-block extremal formula problems/integer_sequences/E0856/claims Sequences and Densities of Integers divisors/erdos_1970_extremal_problems_combinatorial_number_theory Result on p. 127: a reciprocal sum above c log x forces k members with pairwise the same least common multiple Sequences and Densities of Integers integer_sequences/tang_2025_harmonic_lcm_patterns_sunflower_free_capacity Corollary 1.7 (p. 3): (log N)^(log 1.551 - o(1)) <= f_3(N) << (log N)^(3/2^(2/3) - 1 + o(1))Theorem 1.2 (p. 2): an LCM-k-free set in [N] with harmonic sum (log N)^(c_k - o(1))Theorem 1.4 (p. 3): the sunflower-free capacity is 2 exactly when f_k(N) = (log N)^(1 - o(1))Theorem 1.5 (p. 3): f_k(N) << (log N)^(mu_k^S - 1 + o(1))Theorem 1.6 (p. 3): f_k(N) >= (log N)^(log mu_k^S - o(1))

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