Delta-system lemma

Every uncountable family of finite sets has an uncountable subfamily whose distinct members have the same pairwise intersection, called the root. It is also known as the sunflower lemma.

If $\mu <\theta =\mathrm{cf}(\theta )$ are infinite cardinals and $|[\alpha {]}^{<\mu }|<\theta$ for every $\alpha <\theta$, every $\theta$-sized family of sets of size less than $\mu$ has a $\theta$-sized delta-system subfamily.

A family of sets is a Delta-system with root $r$ when any two distinct members intersect exactly in $r$. The members themselves need not be finite, although the usual Delta-system lemma applies to uncountable families of finite sets.

A sunflower is a delta-system: its pairwise intersections all equal one core, and its petals outside the core are pairwise disjoint. For an $s$-uniform set family, prove the displayed bound by induction on $s$. If a maximal disjoint subfamily has $r$ members, their core is empty. Otherwise its union has at most $s(r-1)$ points and meets every member. Some point belongs to more than $(s-1)!(r-1{)}^{s-1}$ members. Remove it and apply the induction hypothesis; putting it back enlarges the common core. The case $s=0$ is immediate.

If $\kappa$ is a regular uncountable cardinal number, every family of $\kappa$ finite sets has a subfamily of cardinality $\kappa$ whose distinct members have one common pairwise intersection.