  How many graphs on an $n$-point set can we find such that any two have
connected intersection? Berger, Berkowitz, Devlin, Doppelt, Durham, Murthy and
Vemuri showed that the maximum is exactly $1/2^{n-1}$ of all graphs. Our aim in
this short note is to give a 'directed' version of this result; we show that a
family of oriented graphs such that any two have strongly-connected
intersection has size at most $1/3^n$ of all oriented graphs. We also show that
a family of graphs such that any two have Hamiltonian intersection has size at
most $1/2^n$ of all graphs, verifying a conjecture of the above authors.
