TITLE: A reference to infinite version of the Sunflower Lemma
QUESTION [9 upvotes]: Please help me to find a proper reference to the following infinite version of the Sunflower Lemma.

Lemma. Let $n\in\mathbb N$. Every infinite family of $n$-element sets contains an infinite subfamily $\mathcal F$ such that $A\cap B=\bigcap\mathcal F$ for any distinct sets $A,B\in\mathcal F$.

Browsing through Internet I could find only finite and uncountable versions of the Sunflower Lemma (but not the countable one).

REPLY [13 votes]: I found it in the book Komjáth, Péter; Totik, Vilmos, Problems and theorems in classical set theory, Problem Books in Mathematics. New York, NY: Springer (ISBN 0-387-30293-X/hbk). xii, 514 p. (2006). ZBL1103.03041. It's stated on p. 107 as the first item in the chapter on $\Delta$-systems:


An infinite family of $n$-element sets ($n\lt\omega$) includes an infinite $\Delta$-subfamily.


A proof is given on p. 421. No reference is given.