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.
Δ lemma for 
If is an -sized collection of countable subsets of , and if CH holds, then there is an -sized Δ-subsystem. Let enumerate . For , let . By Fodor's lemma, fix stationary in such that is constantly equal to on . Build of cardinality such that whenever are in then . Using Ch, there are only -many countable subsets of , 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!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.
- Deza, M.; Frankl, P. (1981), "Every large set of equidistant (0,+1,–1)-vectors forms a sunflower", Combinatorica 1 (3): 225–231, doi:10.1007/BF02579328, ISSN 0209-9683, MR 637827
- Erdős, Paul; Rado, R. (1960), "Intersection theorems for systems of sets", Journal of the London Mathematical Society, Second Series 35 (1): 85–90, doi:10.1112/jlms/s1-35.1.85, ISSN 0024-6107, MR 0111692
- Jech, Thomas (2003). Set Theory. Springer.
- Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. North-Holland. ISBN 0-444-85401-0.