Sunflower Lemma

The Sunflower Lemma, also known as the delta-system lemma, was proved by Paul Erdős and Richard Rado in 1960. It states that for any positive integers k and r there is a bound f(k, r) such that any family of more than f(k, r) sets, each of size k, must contain a sunflower of r sets, meaning r sets whose pairwise intersections are all identical to a single common kernel. Erdős and Rado showed f(k, r) is at most k! times (r minus 1) to the k, and the lemma is a basic tool in extremal combinatorics and in complexity theory circuit lower bounds, while the question of whether the bound can be improved to an exponential one, the sunflower conjecture, remains open.