TITLE: the delta system lemma outside set theory
QUESTION [11 upvotes]: The lemma:
Any uncountable set $S$ of finite sets has an uncountable subset $\Delta \subseteq S$ and an $x$ such that $\forall a,b \in \Delta$, if $a \neq b$ then $a \cap b = x$. $\Delta$ is called a $\Delta$-system.
I've seen this lemma used in independence proofs, such as the famous result that the negation of the continuum hypothesis is consistent with ZFC, but it seems like it would be useful in other fields as well. Does anyone have any examples with this lemma outside pure set theory?
See also the finite and generalized infinite versions posted below.

REPLY [3 votes]: Joel gave one generalization; there is another generalization to a different direction:
Let $T$ be a $\xi$-stable (complete) first order theory, and $A$ a set in a model of $T$ with $|A|\le\xi$. If $I$ has cardinality $>\xi$, then $I$ has a subset with cardinality $>\xi$, which is an indiscernible set over $A$. (Here $I$ is a set of finite tuples in some model of $T$ that contains $A$.)
The usual $\Delta$-lemma follows from this because the theory of infinite set with empty language is $\omega$-stable, and an indiscernible set in such a model is just as wanted.
I believe this is from Saharon Shelah's Classification Theory (1990).
Wikipedia links:

Stable theory
Indiscernibles