Sunflowers

A sunflower with $k$ petals and core $Y$ is a collection of sets $\left\{ F_{1},\ldots ,F_{k}\right\}$ such that $\left|F_{i} \cap F_{j}\right| = Y$ for $i \neq j$ and all the sets $F_{i} \setminus Y$ are nonempty. For example,

• pairwise disjoint sets form a sunflower with an empty core;

• one-dimensional vector subspaces (Sec. 7.1) form a sunflower with a one-element core;

• the sets $X_{p}$ of all rational fractions with a fixed prime denominator $p$ form a sunflower with an infinite core.

Most of the problems in this section are taken from the book [J].

A weak $\Delta$-system is a collection of sets $S_{1},\ldots ,S_{k}$ such that the numbers $\left|S_{i} \cap S_{j}\right|$ are the same for all $i \neq j$. Obviously, a sunflower is a weak $\Delta$-system. The converse is false. However, the following theorem holds.

Deza’s theorem. If $\mathscr {F}$ is a weak $\Delta$-system of $s$-element sets and $\left|\mathscr {F}\right| \geqslant s^{2}-s+2$, then $\mathscr {F}$ is a sunflower.

The proof of the theorem is rather difficult.

Let us show that the above bound is sharp.

The common part of sets $S_{1},\ldots ,S_{k}$ is the union $\bigcup_{i \neq j}(S_{i} \cap S_{j})$ of all their pairwise intersections.

A flower with $k$ petals and core $Y$ is a collection $\mathscr {F}$ of sets such that each of them contains $Y$ and there is no SCR of $k-1$ elements for the collection $\left\{ S \setminus Y : S \in \mathscr {F}\right\}$.