  We study the problem of determining the size of the largest intersecting
$P$-free family for a given partially ordered set (poset) $P$. In particular,
we find the exact size of the largest intersecting $B$-free family where $B$ is
the butterfly poset and classify the cases of equality. The proof uses a new
generalization of the partition method of Griggs, Li and Lu. We also prove
generalizations of two well-known inequalities of Bollob\'{a}s and Greene,
Katona and Kleitman in this case. Furthermore, we obtain a general bound on the
size of the largest intersecting $P$-free family, which is sharp for an
infinite class of posets originally considered by Burcsi and Nagy, when $n$ is
odd. Finally, we give a new proof of the bound on the maximum size of an
intersecting $k$-Sperner family and determine the cases of equality.
