On generalized Turan results in height two posets

In this paper, we show that, for all $ngeq 5$, the maximum number of $2$-chains in a butterfly-free family in the $n$-dimensional Boolean lattice is $leftlceilfrac{n}{2}rightrceilbinom{n}{lfloor n/2rfloor}$. In addition, for the height-2 poset $K_{s,t}$, we show that, for fixed $s$ and $t$, a $K_{s,t}$-free family in the $n$-dimensional Boolean lattice has $Oleft(nbinom{n}{lfloor n/2rfloor}right)$ $2$-chains.