The sunflower conjecture, also called the Erdős-Rado sunflower conjecture, asserts that the maximum cardinality 
of a family
of distinct 
-element
sets containing no 
-member sunflower satisfies
 |
(1)
|
for every integer 
, with a constant 
depending only on the fixed integer 
(Erdős and Rado 1960, p. 86).
The size of the ground set is unrestricted. The conjecture
remains open, even for 
. It is problem
20 in the Erdős problems collection.
The elementary bounds are
 |
(2)
|
The upper bound is the sunflower lemma (Erdős and Rado 1960, Rao 2020), where 
is a factorial. For the lower bound, selecting one element
from each of 
pairwise disjoint sets of size 
gives 
distinct 
-element sets containing no 
-member sunflower (Rao 2020,
p. 1).
Building on the work of Alweiss et al. (2021) and Rao (2020), Bell et al. (2021) proved
 |
(3)
|
where 
is an absolute constant, for integers 
and 
. Here 
is the natural logarithm.
The factor 
still depends on 
, so this bound
does not establish the conjectured 
bound.
The Erdős-Szemerédi sunflower conjecture instead concerns families of subsets of a fixed ground set, without requiring all members to have the same cardinality. Its resolution does not settle the uniform conjecture above (Bucić et al. 2026).
