Induced and Non-induced Forbidden Subposet Problems

The problem of determining the maximum size La(n, P ) that a P -free subposet of the Boolean lattice Bn can have, attracted the attention of many researchers, but little is known about the induced version of these problems. In this paper we determine the asymptotic behavior of La∗(n, P ), the maximum size that an induced P -free subposet of the Boolean lattice Bn can have for the case when P is the complete two-level poset Kr,t or the complete multi-level poset Kr,s1,...,sj ,t when all si’s either equal 4 or are large enough and satisfy an extra condition. We also show lower and upper bounds for the non-induced problem in the case when P is the complete three-level poset Kr,s,t. These bounds determine the asymptotics of La(n,Kr,s,t) for some values of s independently of the values of r and t.