Then there is some such that
and is a Delta System.
Proof
WLOG assume that there is some such that every has .
Now we go by induction on .
For , we get that is a System with root .
Assume the result for , we prove it for .
If some has many with ,
then remove it from everywhere and apply induction hypothesis.
So assume that every has many with .
Order .
Then just build a chain of nonintersecting elements and use Cofinality.
Now the limit case.
Assume the result for all .
idk man does it even matter atp
Special case
Let and .
Every uncountable family of finite sets contains an uncountable Delta System.