In mathematics, a sunflower or Δ system is a collection of sets whose pairwise intersection is constant, and called the kernel.

The Δ-lemma is a combinatorial

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 set-theoretic tool used in proofs to impose an upper bound

Upper bound

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...

 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 ZFC that the continuum hypothesis

Continuum hypothesis

In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...

does not hold.

Formal definition

A Δ-system W is a collection of sets whose pairwise intersection is constant. That is, there exists a fixed S (possibly empty) such that for all A, B ∈ W with A ≠ B, A ∩ B = S.

The Δ-lemma states that every uncountable collection of finite sets contains an uncountable Δ-system.

Sunflower lemma and conjecture

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 that the factor of b! can be replaced by C^b for some constant C.