Induced and non-induced forbidden subposet problems

TL;DR

This paper investigates the maximum size of induced and non-induced forbidden subposet problems within Boolean lattices, providing asymptotic results for specific classes of posets and establishing bounds for others.

Contribution

It determines the asymptotic behavior of induced forbidden subposet problems for complete two-level and multi-level posets, and provides bounds for the non-induced case with three-level posets.

Findings

Asymptotic behavior of $La^*(n,P)$ for certain posets determined.

Bounds established for $La(n,K_{r,s,t})$ in the non-induced case.

Results apply to complete two-level, multi-level, and three-level posets.

Abstract

The problem of determining the maximum size $La(n,P)$ that a $P$-free subposet of the Boolean lattice $B_n$ 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 $B_n$ can have for the case when $P$ is the complete two-level poset $K_{r,t}$ or the complete multi-level poset $K_{r,s_1,\dots,s_j,t}$ when all $s_i$'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 $K_{r,s,t}$. These bounds determine the asymptotics of $La(n,K_{r,s,t})$ for some values of $s$ independently of the values of $r$ and $t$.