TITLE: Sunflowers in maximal almost disjoint families
QUESTION [8 upvotes]: Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say ${\cal A}\subseteq [\omega]^\omega$ is almost disjoint if $A \cap B$ is finite whenever $A\neq B \in {\cal A}$. Zorn's Lemma implies that every almost disjoint family is contained in a maximal one. Moreover, a diagonalization argument shows that every maximal almost disjoint (MAD) family is uncountable.
A sunflower is a set ${\cal X}$ of sets such that ${\cal X} \neq \emptyset$ and there is $K\subseteq \bigcup{\cal X}$ such that $X\cap Y = K$ whenever $X\neq Y\in {\cal X}$. (We allow for $K=\emptyset$.)
Does every MAD family ${\cal A}\subseteq [\omega]^\omega$ contain an infinite sunflower? If not, is it true that given $n\in \omega$, every MAD family contains a sunflower of cardinality $n$?

REPLY [4 votes]: One remark on KP Hart's answer: if one does the same to a finitely branching tree with unbounded sizes of the splitting sets then one gets a MAD family with no infinite delta-system but delta systems of all finite sizes.