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 on a fixed ground set, without fixing their member cardinalities. Its resolution does not settle the uniform sunflower conjecture of Erdős and Rado (Bucić et al. 2026).
