The Brown-Erdős-Sós conjecture asserts that for each integer 
and real number 
, every 3-uniform hypergraph
on sufficiently many vertices 
with at least 
hyperedges contains
distinct hyperedges whose union
has at most 
vertices (Brown et al. 1973).
The case 
is the Ruzsa-Szemerédi theorem.
The conjecture remains open in general (Conlon et al. 2023, Shapira and Tyomkyn
2025). The corresponding 
-uniform formulation for 
replaces the vertex bound by 
.
This is problem 1178 in the Erdős problems collection (Bloom 2026).
-Graphs."
In New Directions in the Theory of Graphs (Ed. F. Harary). New York:
Academic Press, pp. 53-63, 1973.Conlon, D.; Gishboliner, L.; Levanzov,
Y.; and Shapira, A. "A New Bound for the Brown-Erdős-Sós Problem."
J. Combin. Th. Ser. B 158, 1-35, 2023.