On an extremal problem for poset dimension

Let $f(n)$ be the largest integer such that every poset on $n$ elements has a $2$-dimensional subposet on $f(n)$ elements. What is the asymptotics of $f(n)$? It is easy to see that $f(n)\geqslant n^{1/2}$. We improve the best known upper bound and show $f(n)=\mathcal{O}(n^{2/3})$. For higher dimensions, we show $f_d(n)=\mathcal{O}\left(n^\frac{d}{d+1}\right)$, where $f_d(n)$ is the largest integer such that every poset on $n$ elements has a $d$-dimensional subposet on $f_d(n)$ elements.Comment: removed proof of Theorem 3 duplicating previous work; fixed typos and reference

Forbidden subposet problems in the grid

For posets $P$ and $Q$, extremal and saturation problems about weak and strong $P$-free subposets of $Q$ have been studied mostly in the case $Q$ is the Boolean poset $Q_n$, the poset of all subsets of an $n$-element set ordered by inclusion. In this paper, we study some instances of the problem with $Q$ being the grid, and its connections to the Boolean case and to the forbidden submatrix problem

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-

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 $t$

Supersaturation and stability for forbidden subposet problems

We address a supersaturation problem in the context of forbidden subposets. A family $\mathcal{F}$ of sets is said to contain the poset $P$ if there is an injection $i:P \rightarrow \mathcal{F}$ such that $p \le_P q$ implies $i(p) \subset i (q)$. The poset on four elements $a,b,c,d$ with $a,b \le c,d$ is called butterfly. The maximum size of a family $\mathcal{F} \subseteq 2^{[n]}$ that does not contain a butterfly is $\Sigma(n,2)=\binom{n}{\lfloor n/2 \rfloor}+\binom{n}{\lfloor n/2 \rfloor+1}$ as proved by De Bonis, Katona, and Swanepoel. We prove that if $\mathcal{F} \subseteq 2^{[n]}$ contains $\Sigma(n,2)+E$ sets, then it has to contain at least $(1-o(1))E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2}$ copies of the butterfly provided $E\le 2^{n^{1-\varepsilon}}$ for some positive $\varepsilon$. We show by a construction that this is asymptotically tight and for small values of $E$ we show that the minimum number of butterflies contained in $\mathcal{F}$ is exactly $E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2}$