The quadratic Brown--Erdős--Sós problem for 3-uniform hypergraphs with 8 and 9 edges
Abstract
The famous and actively studied problem of Brown--Erdős--Sós from 1973 asks for f(r)(n;s,k), the maximum number of edges in an r-graph with n vertices in which no s vertices span k or more edges. In this paper, we concentrate on the case r=3 and s=k+2, with k≥2 fixed and n→∞; then it is easy to show that the extremal function grows quadratically in n. Delcourt and Postle proved that the limit π(k):=limn→∞ f(3)(n;k+2,k)/n2 exists for every k. While Brown, Erdős and Sós observed that π(2)=1/6 already in the 1970s, the value of π(k) for 3≤ k≤ 7 was determined only recently (by various subgroups of Glock, Joos, Kim, Kühn, Lichev, Pikhurko, and Sun). Very recently, Chao, Huang and Liu determined π(k) for every odd k. Independently of the last result, we show that π(9)=1/5. Also, we prove that π(8)≤ 5053/26544, which is within $0.0029$ of the best known lower bound π(8)≥ 3/16. The new upper bounds are obtained by expressing some previous arguments as a linear program and then using a computer to generate and solve its instances. Our proof of the lower bound on π(9) is based on a finite field construction combined with existing packing results.
Related
- A near-quadratic lower bound for the Brown--Erdős--Sós (9,6)-problem
- Towards the Erdős matching conjecture for 4‐uniform hypergraphs: Stability and applications
- Quadratic bounds for intersecting hypergraphs: An affirmative answer to a question of Duke and Erdős
- The Erdős Matching Conjecture for 4-uniform hypergraphs
- Asymptotics of the Brown--Erdős--Sós problem at integer exponents
BibTeX
@misc{bodnar2026quadratic,
title = {{The quadratic Brown--Erdős--Sós problem for 3-uniform hypergraphs with 8 and 9 edges}},
author = {Levente Bodnár and Oleg Pikhurko and Shumin Sun and Yan Wang and Jiasheng Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03323},
eprint = {2610.03323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03323},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The quadratic Brown--Erdős--Sós problem for 3-uniform hypergraphs with 8 and 9 edges
AU - Levente Bodnár
AU - Oleg Pikhurko
AU - Shumin Sun
AU - Yan Wang
AU - Jiasheng Zeng
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03323
UR - https://arxiv.org/abs/2610.03323
AB - The famous and actively studied problem of Brown--Erdős--Sós from 1973 asks for f^(r)(n;s,k), the maximum number of edges in an r-graph with n vertices in which no s vertices span k or more edges. In this paper, we concentrate on the case r=3 and s=k+2, with k≥2 fixed and n→∞; then it is easy to show that the extremal function grows quadratically in n. Delcourt and Postle proved that the limit π(k):=lim_n→∞ f^(3)(n;k+2,k)/n^2 exists for every k. While Brown, Erdős and Sós observed that π(2)=1/6 already in the 1970s, the value of π(k) for 3≤ k≤ 7 was determined only recently (by various subgroups of Glock, Joos, Kim, Kühn, Lichev, Pikhurko, and Sun). Very recently, Chao, Huang and Liu determined π(k) for every odd k. Independently of the last result, we show that π(9)=1/5. Also, we prove that π(8)≤ 5053/26544, which is within $0.0029$ of the best known lower bound π(8)≥ 3/16. The new upper bounds are obtained by expressing some previous arguments as a linear program and then using a computer to generate and solve its instances. Our proof of the lower bound on π(9) is based on a finite field construction combined with existing packing results.
ER -