The existence of sunflowers in any sufficiently large set is an example of a fascinating field of combinatorics called Ramsey theory. The idea is that for a very wide range of structures, sufficiently large objects always necessarily contain substructures of arbitrary size because there's no way to avoid it as the amount of space inside the object grows.

The classic example is the "Theorem on Friends and Strangers", which states that in any group of six or more people, there is always some subset of three people who are either pairwise friends or pairwise strangers (or, in graph theoretic terms, three vertices that are either all pairwise connected by an edge or pairwise not connected by an edge).

This generalizes to larger subsets: in any group of 18 or more people, there's always some subset of four people who are pairwise friends or strangers, and in any group of 49 or more, there's always some subset of five (49 may not be optimal here; it's known to be between 43 and 49). The known bounds are quite weak: the number of people required to guarantee a mutual-friends-or-strangers subset of n people is known only to be omega(2^(n/2)) and little-o(2^2n), with no improvements on those bounds made despite nearly a century of work.