Given a finite poset $\mathcal P$, we call a family $\mathcal F$ of subsets of $[n]$ $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturated number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we are mainly interested in the four-point poset called the butterfly. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan showed that the saturation number for the butterfly lies between $\log_2{n}$ and $n^2$. We give a linear lower bound of $n+1$. We also prove some other results about the butterfly and the poset $\mathcal N$.
We introduce a method of decomposing the family of intervals along a cyclic permutation into chains to determine the size of the largest family of subsets of $[n]:= \{1,2,...,n\}$ not containing one or more given posets as a subposet. De Bonis, Katona and Swanepoel determined the size of the largest butterfly-free family. We strengthen this result by showing that, for certain posets containing the butterfly poset as a subposet, the same bound holds. We also obtain the corresponding LYM-type inequalities.
In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the Erdős-Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we allow the…
Given a finite poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is induced-$\mathcal{P}$-saturated if $\mathcal{F}$ contains no copy of $\mathcal{P}$ as an induced subposet but every proper superset of $\mathcal{F}$ contains a copy of $\mathcal{P}$ as an induced subposet. The minimum size of an induced-$\mathcal{P}$-saturated family in the $n$-dimensional Boolean lattice, denoted $\operatorname{sat}^*(n,\mathcal{P})$, was first studied by Ferrara et al. (2017). Our work focuses on strengthening lower bounds. For the 4-point poset known as the diamond, we prove $\o…
Given a poset $P$, a family $F$ of elements in the Boolean lattice is said to be $P$-saturated if (1) $F$ contains no copy of $P$ as a subposet and (2) every proper superset of $F$ contains a copy of $P$ as a subposet. The maximum size of a $P$-saturated family is denoted by $La(n,P)$, which has been studied for a number of choices of $P$. The minimum size of a $P$-saturated family, $sat(n,P)$, was introduced by Gerbner et al. (2013), and parallels the deep literature on the saturation function for graphs. We introduce and study the concept of saturation for induced subposets. As opposed to in…
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 $Σ(n,2)=\binom{n}{\lfloor n/2 \rfloor}+\binom{n}{\lfloor n/2 \rfloor+1}$ as proved by De Bonis, Katona, and Swanepoel. We prove that if $\mathcal{F} \subseteq 2^{[n]}$ contains $…
A subfamily $\mathcal{G}\subseteq \mathcal{F}\subseteq 2^{[n]}$ of sets is a non-induced (weak) copy of a poset $P$ in $\mathcal{F}$ if there exists a bijection $i:P\rightarrow \mathcal{G}$ such that $p\le_P q$ implies $i(p)\subseteq i(q)$. In the case where in addition $p\le_P q$ holds if and only if $i(p)\subseteq i(q)$, then $\mathcal{G}$ is an induced (strong) copy of $P$ in $\mathcal{F}$. We consider the minimum number $sat(n,P)$ [resp.\ $sat^*(n,P)$] of sets that a family $\mathcal{F}\subseteq 2^{[n]}$ can have without containing a non-induced [induced] copy of $P$ and being maximal with…
For given posets $P$ and $Q$ and an integer $n$, the generalized Turán problem for posets, asks for the maximum number of copies of $Q$ in a $P$-free subset of the $n$-dimensional Boolean lattice, $2^{[n]}$. In this paper, among other results, we show the following: (i) For every $n\geq 5$, the maximum number of $2$-chains in a butterfly-free subfamily of $2^{[n]}$ is $\left\lceil\frac{n}{2}\right\rceil\binom{n}{\lfloor n/2\rfloor}$. (ii) For every fixed $s$, $t$ and $k$, a $K_{s,t}$-free family in $2^{[n]}$ has $O\left(n\binom{n}{\lfloor n/2\rfloor}\right)$ $k$-chains. (iii) For every $n\geq …
Let $n$ be a positive integer, $q$ a power of a prime, and $\mathcal{L}_n(q)$ the poset of subspaces of an $n$-dimensional vector space over a field with $q$ elements. This poset is a normalized matching poset and the set of subspaces of dimension $\left\lfloor n/2 \right\rfloor$ or those of dimension $\left\lceil n/2 \right\rceil$ are the only maximum-sized anti-chains in this poset. Strengthening this well-known and celebrated result, we show that, except in the case of $\mathcal{L}_3(2)$, these same collections of subspaces are the only maximum-sized families in $\mathcal{L}_n(q)$ that avoi…
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.
We study the problem of determining the size of the largest intersecting $P$-free family for a given partially ordered set (poset) $P$. In particular, we find the exact size of the largest intersecting $B$-free family where $B$ is the butterfly poset and classify the cases of equality. The proof uses a new generalization of the partition method of Griggs, Li and Lu. We also prove generalizations of two well-known inequalities of Bollobás and Greene, Katona and Kleitman in this case. Furthermore, we obtain a general bound on the size of the largest intersecting $P$-free family, which is sharp f…
In this paper we introduce a problem that bridges forbidden subposet and forbidden subconfiguration problems. The sets $F_1,F_2, \dots,F_{|P|}$ form a copy of a poset $P$, if there exists a bijection $i:P\rightarrow \{F_1,F_2, \dots,F_{|P|}\}$ such that for any $p,p'\in P$ the relation $p<_P p'$ implies $i(p)\subsetneq i(p')$. A family $\mathcal{F}$ of sets is \textit{$P$-free} if it does not contain any copy of $P$. The trace of a family $\mathcal{F}$ on a set $X$ is $\mathcal{F}|_X:=\{F\cap X: F\in \mathcal{F}\}$. We introduce the following notions: $\mathcal{F}\subseteq 2^{[n]}$ is $l$-trac…
