Sunflower (mathematics)


In mathematics, a sunflower or -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 ? The -lemma, sunflower lemma, and sunflower conjecture give various conditions that imply the existence of a large sunflower in a given collection of sets.

Formal definition

Suppose is a universe set and is a collection of subsets of. The collection is a sunflower if there is a subset of such that for each distinct and in, we have. In other words, is a sunflower if the pairwise intersection of each set in is constant.
Note that this intersection,, may be empty; a collection of disjoint subsets is also a sunflower.

Δ-lemma

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

Δ-lemma for ''ω''2

If is an -sized collection of countable subsets of, and if the continuum hypothesis holds, then there is an -sized -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

proved the sunflower lemma, stating that if and are positive integers then a collection of sets of cardinality at most contains a sunflower with more than sets.
The sunflower conjecture is one of several variations of the conjecture of that the factor of can be replaced by for some constant. A 2020 paper by Alweiss, Lovett, Wu, and Zhang gives the best progress towards the conjecture, proving the result for .