On Generalized Turán Results in Height Two Posets

For given posets $P$ and $Q$ and an integer $n$, the generalized Turán problem for posets asks for the maximum number of copies of $Q$ in a $P$-free subset of the $n$-dimensional Boolean lattice, $2^{[n]}$. In this paper, among other results, we show the following: (i) For every $n\geq 5$, the maximum number of 2-chains in a butterfly-free subfamily of $2^{[n]}$ is $\lceil\frac{n}{2} \rceil\binom{n}{\lfloor n/2\rfloor}$. (ii) For every fixed $s$, $t$ and $k$, a $K_{s,t}$-free family in $2^{[n]}$ has $O (n\binom{n}{\lfloor n/2\rfloor})$ $k$-chains. (iii) For every $n\geq 3$, the maximum number of $2$-chains in an ${N}$-free family is $\binom{n}{\lfloor n/2\rfloor}$, where ${N}$ is a poset on 4 distinct elements $\{p_1,p_2,q_1,q_2\}$ for which $p_1 < q_1$, $p_2 < q_1$ and $p_2 < q_2$. (iv) We also prove exact results for the maximum number of 2-chains in a family that has no 5-path and asymptotic estimates for the number of 2-chains in a family with no 6-path.