The Erdős-Szemerédi sunflower conjecture asserts that, for every integer 
,
there is a constant 
such that a family of subsets of ![[n]={1,...,n}](/images/equations/Erdos-SzemerediSunflowerConjecture/Inline3.svg)
containing no 
-member sunflower has at most
members, for 
(Bucić et al. 2026). A sunflower
consists of distinct sets whose pairwise intersections
are equal.
Bucić et al. (2026) proved the quantitative bound
 |
for every integer 
and 
. Together with the elementary strict bound 
when 
, this proves the conjecture. The result concerns families
on a fixed ground set. It does not settle the uniform
sunflower conjecture, which instead bounds families of fixed-size sets
independently of the size of their ground set.
Bucić et al. (2026) report that ChatGPT originated the key vector-sunflower argument. The authors rewrote and checked the proof. They report a Lean verification whose public release was forthcoming.
