  We show that the cliques of maximal size in the confluence graph of an
arbitrary unital of order $q>2$ have size $q^2$, and that these cliques are the
pencils of all blocks through a given point. This solves the Erd\H{o}s-Ko-Rado
problem for all unitals. We also determine all maximal cliques of the
confluence graph of the Hermitian unitals. As an application, we show that the
confluence graph of an arbitrary unital unambiguously determines the unital.
Along the way, we show that each linear space with $q^2$ points such that the
sizes of both point rows and line pencils are bounded above by $q+1$ embeds in
a projective plane of order $q$.
