In mathematics, a sunflower or Δ system is a collection of sets whose pairwise intersection is constant, and called the kernel.
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.
The original term for this concept was "Δ-system". More recently the term "sunflower", possibly introduced by Deza & Frankl (1981), has been gradually replacing it.
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 ZFC that the continuum hypothesis does not hold.
A Δ-system W is a collection of sets whose pairwise intersection is constant. That is, there exists a fixed S called the kernel (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.
Erdős & Rado (1960, p. 86) proved the sunflower lemma, stating that if a and b are positive integers then a collection of b!ab+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 Cb for some constant C.