Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 857

claims/: The 1 claim page of Problem 857, one per claimant's result; the problem's standing derives from them.

Statement. Let $m=m(n,k)$ be minimal such that in any collection of sets $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ there must exist a sunflower of size $k$ - that is, some collection of $k$ of the $A_i$ which pairwise have the same intersection.

Estimate $m(n,k)$ , or even better, give an asymptotic formula.

Status. Open.

Source. erdosproblems.com/857, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #857, https://www.erdosproblems.com/857.

References.

[ASU13] Alon, Noga and Shpilka, Amir and Umans, Christopher, On sunflowers and matrix multiplication. Comput. Complexity (2013), 219-243.
[Er70] Erdős, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133.
[NaSa17] Naslund, Eric and Sawin, Will, Upper bounds for sunflower-free sets. Forum Math. Sigma (2017), Paper No. e15, 10.

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation asks for an estimate, or an asymptotic formula, for $m(n,k)$ , the least $m$ such that any $m$ subsets of $\{1,\ldots,n\}$ contain $k$ with pairwise equal intersections; the site calls this the weak sunflower problem and labels it OPEN. The only bound its commentary credits is for $k=3$ : Naslund and Sawin prove $m(n,3)\le(3/2^{2/3})^{(1+o(1))n}$ , with $3/2^{2/3}=1.889\ldots$ , by the polynomial method, refereed in Forum Math. Sigma (card). That claim is accepted and partial; a bound for one $k$ settles no estimate of $m(n,k)$ , so the problem's standing stays open. The site also notes the connection, observed by Alon, Shpilka and Umans [ASU13] (card), between the case $k=3$ and the cap set problem, the largest subset of $\mathbb F_3^n$ with no three-term arithmetic progression; Naslund and Sawin's Theorem 3 quantifies that reduction as $\mu_3^S\le\sqrt{1+C}$ , $C$ the cap set capacity, which with the Ellenberg-Gijswijt bound $C\le2.7552$ gives only $1.938$ . The site credits no asymptotic formula and no bound for $k\ge4$ . Erdős's 1970 formulation [Er70] (card) asks the equivalent question with unions in place of intersections.

A thread post of 26 February 2026 reports a Lean 4 development (github.com/SproutSeeds/sunflower-lean) that certifies exact values of its weak sunflower numbers $M(n,3)$ for $n\le7$ , among them $M(1,3)=2$ , $M(4,3)=8$ , $M(5,3)=12$ , $M(6,3)=19$ and $M(7,3)=29$ , the cases $n\le4$ by decision in Lean and the rest through SAT solving with checked LRAT certificates; the post says the development was carried out with OpenAI Codex and Anthropic Claude generating candidate proofs, with one exploratory call to Aristotle. It is a thread post linking a repository, not a dated manuscript, and exact values at $n\le7$ settle no part of the asymptotic question, so it has no claim page; this corpus has not built or audited the development.

Search scope, 2026-10-07: the site's page and discussion thread (one comment, no proof claims), the community database (teorth/erdosproblems, which lists the problem as open with a formalized statement), the formal-conjectures statement file (857.lean, which leaves the asymptotic answer as sorry and names no formal proof), the journal and arXiv records of Naslund and Sawin's paper, and the linked library cards. No other bound on $m(n,k)$ credited by the site or found in these sources is recorded.

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 (24)

Problem 856 Polylogarithmic bounds tied to the sunflower-free capacity RayYoung, Zhu and Luo's exact exponent by a finite-block extremal formula Naslund and Sawin's bound for three-sunflower-free families problems/set_systems/E0857/claims Set Systems, Designs and Hypergraphs 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 integer_sequences/tang_2025_harmonic_lcm_patterns_sunflower_free_capacity Theorem 1.4 (p. 3): the sunflower-free capacity is 2 exactly when f_k(N) = (log N)^(1 - o(1))number_theory/guy_1991_western_number_theory_problems Problem 91:01 (p. 9): three integers with pairwise the same least common multiple in a set of positive density Set Systems, Designs and Hypergraphs set_systems/alon_2013_sunflowers_matrix_multiplication Theorem 2.3 (p. 3): the uniform 3-sunflower bound c^s implies the {0,1}^n conjecture Theorem 2.7 (p. 6): the {0,1}^n sunflower conjecture is equivalent to the weak one in Z_D^n Theorem 3.2 (p. 9): the {0,1}^n sunflower conjecture bounds Coppersmith-Winograd sets Theorem 3.7 (p. 10): strong USPs give multicolored-sunflower-free collections Theorem 3.9 (p. 13): a strong USP capacity gap implies the weak sunflower conjecture in Z_D^n set_systems/alweiss_2020_improved_bounds_sunflower_lemma Theorem 4.1: a family of at least 2^{n(1-c/log n)} subsets of an n-set contains an r-sunflower set_systems/naslund_2017_upper_bounds_sunflower_free_sets Theorem 3 (p. 2): a sunflower-free family of subsets of [n] has at most 3(n+1) sum_{k<=n/3} C(n,k) members Theorem 8 (p. 4): the sunflower-free capacity is at most sqrt(1+C), C the cap set capacity

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