Delta systems and roots

A family $\mathcal{F}$ of sets is a delta system with root $r$ if $a\cap b=r$ for all distinct $a,b\in \mathcal{F}$, using intersection as in The intersection $\bigcap x$ of a nonempty set, the binary intersection $a\cap b:=\bigcap \{a,b\}$, and disjointness. A finite-set delta system additionally requires each member to be finite; the family itself may be infinite.

For a family indexed by a set $I$, the indexed delta-system condition with root $r$ is $a_i\cap a_j=r$ for all distinct $i,j\in I$. Repetition of sets is permitted in this indexed convention. In particular if two distinct indices have the same value $a$, their intersection condition forces $r=a$.

No additional containment condition on $r$ is imposed for families with fewer than two members: their pairwise condition is vacuous, so they may be assigned any root. For a family with at least two members the root is their common pairwise intersection and is unique. Pairwise disjoint members have root $\mathrm{\varnothing }$. None of these definitions asserts that a large delta subsystem exists.