# 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 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.

## Δ lemma

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. It was introduced by Shanin (1946).

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, if AB then A ∩ B = S.

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

## Δ lemma for $\omega_2$

If $W$ is an $\omega_2$-sized collection of countable subsets of $\omega_2$, and if CH holds, then there is an $\omega_2$-sized Δ-subsystem. Let $\langle A_\alpha:\alpha<\omega_2\rangle$ enumerate $W$. For ${\rm cf}(\alpha)=\omega_1$, let $f(\alpha)={\rm sup}(A_\alpha\cap\alpha)$. By Fodor's lemma, fix $S$ stationary in $\omega_2$ such that $f$ is constantly equal to $\beta$ on $S$. Build $S'\subseteq S$ of cardinality $\omega_2$ such that whenever $i are in $S'$ then $A_i\subseteq j$. Using CH, there are only $\omega_1$-many countable subsets of $\beta$, 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 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.