Intersecting $P$-free families

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.