The maximum size, $La(n,P)$, of a family of subsets of $[n]=\{ 1,2,...,n\}$ without containing a copy of $P$ as a subposet, has been intensively studied. Let $P$ be a graded poset. We say that a family $\mathcal{ F}$ of subsets of $[n]=\{ 1,2,...,n\}$ contains a \emph{ rank-preserving} copy of $P$ if it contains a copy of $P$ such that elements of $P$ having the same rank are mapped to sets of same size in $\mathcal{ F}$. The largest size of a family of subsets of $[n]=\{ 1,2,...,n\}$ without containing a rank-preserving copy of $P$ as a subposet is denoted by $La_{ rp}(n,P)$. Clearly, $La(n,P) \le La_{ rp}(n,P)$ holds. In this paper we prove asymptotically optimal upper bounds on $La_{ rp}(n,P)$ for tree posets of height $2$ and monotone tree posets of height $3$, strengthening a result of Bukh in these cases. We also obtain the exact value of $La_{ rp}(n,\{ Y_{ h,s},Y_{ h,s}'\})$ and $La(n,\{ Y_{ h,s},Y_{ h,s}'\})$, where $Y_{ h,s}$ denotes the poset on $h+s$ elements $x_1,\dots,x_h,y_1,\dots,y_s$ with $x_1<\dots<x_h<y_1,\dots,y_s$ and $Y'_{ h,s}$ denotes the dual poset of $Y_{ h,s}$.
Categories: math.CO