Related papers: Linear Saturation for $\mathcal N$ via Butterflies

Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $G$ is called $\mathcal{F}$-saturated if it does not contain any members of $\mathcal{F}$ but adding any edge creates a copy of some $r$-graph in $\mathcal{F}$. The saturation number…

Let $G$ be a graph and $\mathcal{F}$ be a family of graphs. We say a graph $G$ is $\mathcal{F}$-saturated if $G$ does not contain any member in $\mathcal{F}$ and for any $e\in E(\overline{G})$, $G+e$ creates a copy of some member in $…

Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in\mathcal{F}$, but the addition of any edge $e\notin E(G)$ creates at least one copy…

A 0-1 matrix $M$ contains a 0-1 matrix $P$ if $M$ has a submatrix $P'$ which can be turned into $P$ by changing some of the ones to zeroes. Matrix $M$ is $P$-saturated if $M$ does not contain $P$, but any matrix $M'$ derived from $M$ by…

Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in\mathcal{F}$ but the addition of any edge $e\notin E(G)$ creates at least one copy of…

For given positive integers $k$ and $n$, a family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is $k$-antichain saturated if it does not contain an antichain of size $k$, but adding any set to $\mathcal{F}$ creates an antichain of size $k$.…

We call a family $\mathcal{F}$ of subsets of $[n]$ $s$-saturated if it contains no $s$ pairwise disjoint sets, and moreover no set can be added to $\mathcal{F}$ while preserving this property (here $[n] = \{1,\ldots,n\}$). More than 40…

The linear saturation number $sat^{lin}_k(n,\mathcal{F})$ (linear extremal number $ex^{lin}_k(n,\mathcal{F})$) of $\mathcal{F}$ is the minimum (maximum) number of hyperedges of an $n$-vertex linear $k$-uniform hypergraph containing no…

In this paper, we discuss a generalization of the notion of saturation in graphs in order to deal with induced structures. In particular, we define ${\rm indsat}(n,H)$, which is the fewest number of gray edges in a trigraph so that no…

In the area of forbidden subposet problems we look for the largest possible size $La(n,P)$ of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain a forbidden inclusion pattern described by $P$. The main conjecture of the area…

A graph $G$ is $H$-saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$. The saturation number, $sat (n, H)$, is the minimum number of edges of a graph in the set of all…

Let $H$ be a fixed graph, a graph G is $H$-saturated if it has no copy of $H$ in $G$, but the addition of any edge in $E(\overline G)$ to $G$ results in an $H$-subgraph. The saturation number sat$(n,H)$ is the minimum number of edges in an…

We consider the problem of determining the maximum number of pairs $F\subseteq F'$ in a family $\mathcal{F}\subseteq 2^{[n]}$ that avoids certain posets $P$ of height 2. We show that for any such $P$ the number of pairs is…

We consider 'supersaturation' problems in partially ordered sets (posets) of the following form. Given a finite poset $P$ and an integer $m$ greater than the cardinality of the largest antichain in $P$, what is the minimum number of…

We develop a powerful tool for embedding any tree poset $P$ of height $k$ in the Boolean lattice which allows us to solve several open problems in the area. We show that: * If $H$ is a family in $B_n$ with $|H|\ge (q-1+\varepsilon){n\choose…

The saturation number of a graph $F$, written $\textup{sat}(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. One of the earliest results on saturation numbers is due to Erd\H{o}s, Hajnal, and Moon who determined…

The saturation number $\text{sat}(n,\mathcal{F})$ is the minimum number of edges in any graph which does not contain a member of $\mathcal{F}$ as a subgraph, but will if any edge is added. We give a few upper and lower bounds for saturation…

Given positive integer $n$ and graph $F$, the saturation number $\mathrm{sat}(n, F)$ is the minimum number of edges in an edge-maximal $F$-free graph on $n$ vertices. In this paper, we determine asymptotic behavior of $\mathrm{sat}(n, F)$…

Bonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study \emph{$1$-saturating} linear sets in PG$(2,q^4)$,…

A 0-1 matrix $M$ contains a 0-1 matrix pattern $P$ if we can obtain $P$ from $M$ by deleting rows and/or columns and turning arbitrary 1-entries into 0s. The saturation function $\mathrm{sat}(P,n)$ for a 0-1 matrix pattern $P$ indicates the…