# Sunflower (mathematics)

A mathematical sunflower can be pictured as a flower. The kernel of the sunflower is the brown part in the middle, and each set of the sunflower is the union of a petal and the kernel.

In mathematics, a sunflower or ${\displaystyle \Delta }$-system[1] 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 (a sunflower with many sets)? The ${\displaystyle \Delta }$-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 ${\displaystyle U}$ is a universe set and ${\displaystyle W}$ is a collection of subsets of ${\displaystyle U}$. The collection ${\displaystyle W}$ is a sunflower (or ${\displaystyle \Delta }$-system) if there is a subset ${\displaystyle S}$ of ${\displaystyle U}$ such that for each distinct ${\displaystyle A}$ and ${\displaystyle B}$ in ${\displaystyle W}$, we have ${\displaystyle A\cap B=S}$. In other words, ${\displaystyle W}$ is a sunflower if the pairwise intersection of each set in ${\displaystyle W}$ is constant.

## Δ-lemma

The ${\displaystyle \Delta }$-lemma states that every uncountable collection of finite sets contains an uncountable ${\displaystyle \Delta }$-system.

The ${\displaystyle \Delta }$-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 ω2

If ${\displaystyle W}$ is an ${\displaystyle \omega _{2}}$-sized collection of countable subsets of ${\displaystyle \omega _{2}}$, and if the continuum hypothesis holds, then there is an ${\displaystyle \omega _{2}}$-sized ${\displaystyle \Delta }$-subsystem. Let ${\displaystyle \langle A_{\alpha }:\alpha <\omega _{2}\rangle }$ enumerate ${\displaystyle W}$. For ${\displaystyle {\rm {cf}}(\alpha )=\omega _{1}}$, let ${\displaystyle f(\alpha )={\rm {sup}}(A_{\alpha }\cap \alpha )}$. By Fodor's lemma, fix ${\displaystyle S}$ stationary in ${\displaystyle \omega _{2}}$ such that ${\displaystyle f}$ is constantly equal to ${\displaystyle \beta }$ on ${\displaystyle S}$. Build ${\displaystyle S'\subseteq S}$ of cardinality ${\displaystyle \omega _{2}}$ such that whenever ${\displaystyle i are in ${\displaystyle S'}$ then ${\displaystyle A_{i}\subseteq j}$. Using the continuum hypothesis, there are only ${\displaystyle \omega _{1}}$-many countable subsets of ${\displaystyle \beta }$, 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 ${\displaystyle a}$ and ${\displaystyle b}$ are positive integers then a collection of ${\displaystyle b!a^{b+1}}$ sets of cardinality at most ${\displaystyle b}$ contains a sunflower with more than ${\displaystyle a}$ sets. The sunflower conjecture is one of several variations of the conjecture of (Erdős & Rado 1960, p. 86) that the factor of ${\displaystyle b!}$ can be replaced by ${\displaystyle C^{b}}$ for some constant ${\displaystyle C}$.

For ${\displaystyle a=2}$ this is now known to be true, with constant ${\displaystyle C<2}$.[2]

## Notes

1. ^ The original term for this concept was "${\displaystyle \Delta }$-system". More recently the term "sunflower", possibly introduced by Deza & Frankl (1981), has been gradually replacing it.
2. ^ Kalai, Gil. "The Erdos-Szemeredi Sunflower Conjecture is Now Proven". Retrieved 13 May 2017.