On generalized Turán results in height two posets

Abstract

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 $n\ge 5$, the maximum number of $2$-chains in a butterfly-free subfamily of $2^{[n]}$ is $\lceil \frac{n}{2}\rceil \left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$. (ii) For every fixed $s$, $t$ and $k$, a $K_{s,t}$-free family in $2^{[n]}$ has $O\left(n\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)\right)$ $k$-chains. (iii) For every $n\ge 3$, the maximum number of $2$-chains in an $\text{N}$-free family is $\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$, where $\text{N}$ 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.

13 pages, 3 figures