In mathematics, a sunflower or
Δ
-system is a collection of sets whose pairwise intersection is constant. This constant intersection is called the kernel of the sunflower.
The main research question related to sunflowers is: under what conditions does there exist a large sunflower (a sunflower with many sets)? The
Δ
-lemma, sunflower lemma, and sunflower conjecture give various conditions that imply the existence of a large sunflower in a given collection of sets.
Suppose
U
is a universe set and
W
is a collection of subsets of
U
. The collection
W
is a sunflower (or
Δ
-system) if there is a subset
S
of
U
such that for each distinct
A
and
B
in
W
, we have
A
∩
B
=
S
. In other words,
W
is a sunflower if the pairwise intersection of each set in
W
is constant.
The
Δ
-lemma states that every uncountable collection of finite sets contains an uncountable
Δ
-system.
The
Δ
-lemma is a combinatorial set-theoretic tool used in proofs to impose an upper bound on the size of a collection of pairwise incompatible elements in a forcing poset. It may for example be used as one of the ingredients in a proof showing that it is consistent with Zermelo-Fraenkel set theory that the continuum hypothesis does not hold. It was introduced by Shanin (1946).
If
W
is an
ω
2
-sized collection of countable subsets of
ω
2
, and if the continuum hypothesis holds, then there is an
ω
2
-sized
Δ
-subsystem. Let
⟨
A
α
:
α
<
ω
2
⟩
enumerate
W
. For
c
f
(
α
)
=
ω
1
, let
f
(
α
)
=
s
u
p
(
A
α
∩
α
)
. By Fodor's lemma, fix
S
stationary in
ω
2
such that
f
is constantly equal to
β
on
S
. Build
S
′
⊆
S
of cardinality
ω
2
such that whenever
i
<
j
are in
S
′
then
A
i
⊆
j
. Using the continuum hypothesis, there are only
ω
1
-many countable subsets of
β
, so by further thinning we may stabilize the kernel.
Sunflower lemma and conjecture
Erdős & Rado (1960, p. 86) proved the sunflower lemma, stating that if
a
and
b
are positive integers then a collection of
b
!
a
b
+
1
sets of cardinality at most
b
contains a sunflower with more than
a
sets. The sunflower conjecture is one of several variations of the conjecture of (Erdős & Rado 1960, p. 86) that the factor of
b
!
can be replaced by
C
b
for some constant
C
.
