A sunflower, also called a delta-system, is a family of at least two distinct sets whose pairwise intersections
are all equal. For members 
,
, ..., 
, there is a set 
such that
 |
(1)
|
whenever 
.
The set 
is the core or kernel, and the sets 
are the petals. The petals are pairwise
disjoint. The core may be empty, so a family of
pairwise disjoint sets is a sunflower. For example,
, 
, and 
form a three-member sunflower with core 
.
A sunflower in this combinatorial sense is distinct from the flower-like figures called daisies and the Vogel spiral models of phyllotaxis.
The sunflower lemma (Erdős and Rado 1960) states that a family 
of distinct 
-element sets contains an 
-member sunflower if
 |
(2)
|
for integers 
and 
, where 
is a factorial. An induction
on 
proves the bound,
with the case 
immediate. A maximal pairwise disjoint subfamily either has at least 
members or has union of size at
most 
meeting every member of 
. In the latter case, some element belongs to more than 
members. Deleting that
element and applying the induction hypothesis produces a sunflower, and restoring
the element adds it to the core.
The sunflower conjecture asks whether, for every fixed 
, a constant 
exists such that 
already forces an 
-member sunflower in every family of distinct 
-element sets (Erdős and Rado
1960). The conjecture remains open.
Let 
be the largest size of a family
of distinct four-element sets with no three-member sunflower.
Babanskyy (2026) reported
 |
(3)
|
The upper bound would imply that every family of at least 69 distinct four-element sets contains a three-member sunflower.
It does not exclude size 68 or determine 
. The proof imports a bound of 27 for intersecting four-element-set
families with no three-member sunflower from Axante et al. (2026) and combines
written reductions with exact finite computations. The author reports OpenAI model
assistance with the mathematics and software. As of Oct. 15, 2026, independent
specialist review of the reductions and imported bound, and independent reproduction
of the complete computations, had not been reported.
The Erdős-Szemerédi sunflower conjecture concerns families of subsets
of an 
-element ground set,
without fixing their cardinalities. Bucić et al. (2026) proved this conjecture
with the explicit threshold 
for a 
-petal sunflower, for every 
. This does not settle the uniform sunflower conjecture
of Erdős and Rado.
