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.
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).
Δ lemma for 
If is an -sized collection of countable subsets of , and if the continuum hypothesis 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 the continuum hypothesis, 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.
- Shanin, N. A. (1946), "A theorem from the general theory of sets", C. R. (Doklady) Acad. Sci. URSS (N.S.) 53: 399–400