Erdős–Rado sunflower lemma ($|\mathcal{F}|>s!(r-1{)}^s⟹\text{an }r\text{-member sunflower}$)

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.