For integers r,k≥ 2 and s≥ r, let f^(r)(n;s,k) be the maximum number of edges in an r-uniform hypergraph H on n vertices such that no set of k distinct edges has union of size at most s. For integers r>t≥ 2 and k≥ 2, determine the exact asymptotic coefficient [ pi(r,t,k):=lim_n→∞n^-tf^(r)bigl(n;(r-t)k+t,k bigr), ] where the limit is known to exist. In particular, the remaining unresolved case is to determine pi(3,2,k) for even k≥ 4.

Brown–Erdős–Sós problem

For integers $r,k\ge 2$ and $s\ge r$ , let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $r$ -uniform hypergraph $H$ on $n$ vertices such that no set of $k$ distinct edges has union of size at most $s$ . For integers $r>t\ge 2$ and $k\ge 2$ , determine the exact asymptotic coefficient

\pi(r,t,k):=\lim_{n\to\infty}n^{-t}f^{(r)}\bigl(n;(r-t)k+t,k\bigr),

where the limit is known to exist. In particular, the remaining unresolved case is to determine $\pi(3,2,k)$ for even $k\ge 4$ .

References

Primary source

arXiv

Additional references

Asymptotics of the Brown--Erdős--Sós problem at integer exponents — arXiv — Ting-Wei Chao, Xinqi Huang, Hong Liu

extremal-combinatorics hypergraph-theory

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A September 2026 advance determines most leading coefficients, but the exceptional even-parameter family remains open.

Brown, Erdős, and Sós posed the problem in 1973: determine asymptotic densities for sparse uniform hypergraphs. The broader limit-existence question is now settled, while determining the exact coefficient remains incomplete.

Known results

Brown, Erdős, and Sós (1973) proved the base case $k=2$ .
Glock (2019) proved the $k=3$ case.
Glock, Joos, Kim, Kühn, Lichev, and Pikhurko (2024) proved $\pi(3,4)=7/36$ .
Delcourt and Postle (2023) proved existence of the limit for every $k\ge 2$ when $r=3$ ; related work extends existence to all uniformities.

September 2026 leading-coefficient advance

Ting-Wei Chao, Xinqi Huang, and Hong Liu claim explicit formulas for odd $k$ and for even $k$ with $r\ge4$ , and prove a coefficient greater than $1/6$ for the remaining even- $k$ , $(r,t)=(3,2)$ family. This is substantial claimed progress, but that exceptional family is not resolved.

Current status (as of September 2026): Limit existence is settled broadly and many exact coefficients are known, but the even- $k$ , $(r,t)=(3,2)$ coefficient problem remains open; the latest formulas are an unverified preprint claim.

Sources

No solutions have been posted yet.