Sunflower (mathematics)
In the mathematical fields of set theory and extremal combinatorics, a sunflower or -system is a collection of sets in which all possible distinct pairs of sets share the same intersection. This common intersection is called the kernel of the sunflower.
For any sunflower size, does every set of uniformly sized sets which is of cardinality greater than some exponential in the set size contain a sunflower?
The naming arises from a visual similarity to the botanical sunflower, arising when a Venn diagram of a sunflower set is arranged in an intuitive way. Suppose the shared elements of a sunflower set are clumped together at the centre of the diagram, and the nonshared elements are distributed in a circular pattern around the shared elements. Then when the Venn diagram is completed, the lobe-shaped subsets, which encircle the common elements and one or more unique elements, take on the appearance of the petals of a flower.
The main research question arising in relation to sunflowers is: under what conditions does there exist a large sunflower (a sunflower with many sets) in a given collection of sets? The -lemma, sunflower lemma, and the Erdős-Rado sunflower conjecture give successively weaker conditions which would imply the existence of a large sunflower in a given collection, with the latter being one of the most famous open problems of extremal combinatorics.