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 $B_n$ 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 $t$

An improvement of the general bound on the largest family of subsets avoiding a subposet

Let $La(n,P)$ be the maximum size of a family of subsets of $[n]= \{1,2, ..., n \}$ not containing $P$ as a (weak) subposet, and let $h(P)$ be the length of a longest chain in $P$. The best known upper bound for $La(n,P)$ in terms of $|P|$ and $h(P)$ is due to Chen and Li, who showed that $La(n,P) \le \frac{1}{m+1} \left(|P| + \frac{1}{2}(m^2 +3m-2)(h(P)-1) -1 \right) {\binom {n} {\lfloor n/2 \rfloor}}$ for any fixed $m \ge 1$. In this paper we show that $La(n,P) \le \frac{1}{2^{k-1}} (|P| + (3k-5)2^{k-2}(h(P)-1) - 1 ) {n \choose {\lfloor n/2\rfloor} }$ for any fixed $k \ge 2$, improving the best known upper bound. By choosing $k$ appropriately, we obtain that $La(n,P) = O\left( h(P) \log_2\left(\frac{|P|}{h(P)}+2\right) \right) {n \choose \lfloor n/2 \rfloor }$ as a corollary, which we show is best possible for general $P$. We also give a different proof of this corollary by using bounds for generalized diamonds. We also show that the Lubell function of a family of subsets of $[n]$ not containing $P$ as an induced subposet is $O(n^c)$ for every $c>\frac{1}{2}$.Comment: Corrected mistakes, improved the writing. Also added a result about the Lubell function with forbidden induced subposets. The final publication will be available at Springer via http://dx.doi.org/10.1007/s11083-016-9390-

Sperner type theorems with excluded subposets

Let F be a family of subsets of an n-element set. Sperner's theorem says that if there is no inclusion among the members of F then the largest family under this condition is the one containing all ⌊ frac(n, 2) ⌋-element subsets. The present paper surveys certain generalizations of this theorem. The maximum size of F is to be found under the condition that a certain configuration is excluded. The configuration here is always described by inclusions. More formally, let P be a poset. The maximum size of a family F which does not contain P as a (not-necessarily induced) subposet is denoted by La (n, P). The paper is based on a lecture of the author at the Jubilee Conference on Discrete Mathematics [Banasthali University, January 11-13, 2009], but it was somewhat updated in December 2010. © 2011 Elsevier B.V. All rights reserved