Sunflower (mathematics)

In mathematics, a sunflower or ${\displaystyle \Delta }$-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 (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.

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.

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