Forbidding Rank-Preserving Copies of a Poset

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 extensively studied. Let P be a graded poset. We say that a family F of subsets of [n] = {1, 2, ..., n} contains a 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 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) <= 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,Y-s, Y-h,Y-s'}) and La(n, {Y-h,Y-s, Y-h,Y-s'}), where Y-h,Y-s denotes the poset on h + s elements x(1), ..., x(h), y(1), ..., y(s) with x(1) < ... < x(h) < y(1), ..., y(s) and Y-h,Y-s' denotes the dual poset of Y-h,Y-s, thereby proving a conjecture of Martin et. al. [10].