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Claim. Problem 857 asks for estimates of $m(n,k)$, the least $m$ such that any $m$ subsets of $\{1,\ldots,n\}$ include $k$ with pairwise equal intersections. Naslund and Sawin prove by the polynomial method that a family of subsets of $\{1,\ldots,n\}$ with no three such sets has at most $3(n+1)\sum_{i\le n/3}\binom{n}{i}$ members (Theorem 1 of the published paper, Theorem 3 of the arXiv version; the abstract prints the factor as $3n$), which is at most $(3/2^{2/3})^{(1+o(1))n}$. Hence $m(n,3)\le(3/2^{2/3})^{(1+o(1))n}$, where $3/2^{2/3}=1.889\ldots$. The source card holds the digest. The proof applies the slice-rank method of Croot, Lev and Pach and of Ellenberg and Gijswijt, in Tao's formulation, directly to a function of three sets that detects a sunflower, after splitting the family by set size.

Covers. The case $k=3$, an upper bound only: $m(n,3)\le(3/2^{2/3})^{(1+o(1))n}$. The paper gives no matching lower bound and notes a large gap between the upper and lower bounds for the capacity, gives no bound for $k\ge4$, and gives no asymptotic formula, which is what the problem asks for. Its Theorem 2, on sunflower-free sets in $(\mathbb Z/D\mathbb Z)^n$, and its Theorem 3, the bound $\sqrt{1+C}$ by the cap set capacity $C$ that quantifies the reduction of Alon, Shpilka and Umans and gives only $1.938$, do not improve the bound on $m(n,3)$.

Depends on. Nothing in this wiki: the proof is self-contained apart from the slice-rank method it cites.

Acceptance. Refereed: Forum Math. Sigma 5 (2017), e15, 10 pages, received 16 July 2016, accepted 3 November 2016 and published 27 June 2017 under the Creative Commons Attribution 4.0 license, after its first posting as arXiv:1606.09575 on 2016-06-30. The site's commentary credits Naslund and Sawin with the bound but labels the problem OPEN, so the curator's mention settles nothing and the page lists no reviewed evidence. The formal-conjectures catalog states the problem with its answer left as sorry and names no formal proof, so the page lists no formalized evidence. The library card summarizes the paper and is not acceptance evidence.