Leírás
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 F of subsets of [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] 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. 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 and exact answer for a problem with two forbidden posets.
(joint work with Dániel Gerbner, Abhishek Methuku, and Balázs Patkós)