  We present equivalent formulations for concepts related to set families for
which every subfamily with empty intersection has a bounded sub-collection with
empty intersection. Hereby, we summarize the progress on the related questions
about the maximum size of such families.
  In this work we solve a boundary case of a problem of Tuza for non-trivial
$q$-Helly families, by applying Karamata's inequality and determining the
minimum size of a $2$-self-centered graph for which the common neighborhood of
every pair of vertices contains a clique of size $q-2$.
