Related papers: The Saturation Number for the Diamond is Linear

Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a…

Given a family of graphs $\mathcal{F}$, we define the $\mathcal{F}$-saturation game as follows. Two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an…

Let $H$ be a fixed graph. We say that a graph $G$ is $H$-saturated if it has no subgraph isomorphic to $H$, but the addition of any edge to $G$ results in an $H$-subgraph. The saturation number $\mathrm{sat}(H,n)$ is the minimum number of…

Let $G$ be a graph and $\mathcal{H}$ be a family of graphs. We say $G$ is $\mathcal{H}$-saturated if $G$ does not contain a copy of $H$ with $H\in\mathcal{H}$, but the addition of any edge $e\notin E(G)$ creates at least one copy of some…

The dimension of a poset $P$, denoted $\dim(P)$, is the least positive integer $d$ for which $P$ is the intersection of $d$ linear extensions of $P$. The maximum dimension of a poset $P$ with $|P|\le 2n+1$ is $n$, provided $n\ge2$, and this…

Let $H$ be a fixed graph. A graph $G$ is called {\it $H$-saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$-subgraph. The {\it saturation number} of $H$, denoted $sat(n,H)$, is the…

We say that an edge-coloring of a graph $G$ is proper if every pair of incident edges receive distinct colors, and is rainbow if no two edges of $G$ receive the same color. Furthermore, given a fixed graph $F$, we say that $G$ is rainbow…

Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et…

A graph $H$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to…

A 0-1 matrix $M$ is saturating for a 0-1 matrix $P$ if $M$ does not contain a submatrix that can be turned into $P$ by flipping any number of its $1$-entries to $0$-entries, and changing any $0$-entry to $1$-entry of $M$ introduces a copy…

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…

In a projective plane $\Pi_{q}$ (not necessarily Desarguesian) of order $q$, a point subset $\mathcal{S}$ is saturating (or dense) if any point of $\Pi_{q}\setminus \mathcal{S}$ is collinear with two points in $\mathcal{S}$. Modifying an…

The weak saturation number $\mathrm{wsat}(n,F)$ is the minimum number of edges in a graph on $n$ vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of $F$. A usual approach to prove a…

Given a family ${\mathcal F}$ and a host graph $H$, a graph $G\subseteq H$ is ${\mathcal F}$-saturated relative to $H$ if no subgraph of $G$ lies in ${\mathcal F}$ but adding any edge from $E(H)-E(G)$ to $G$ creates such a subgraph. In the…

For an edge-ordered graph $G$, we say that an $n$-vertex edge-ordered graph $H$ is $G$-saturated if it is $G$-free and adding any new edge with any new label to $H$ introduces a copy of $G$. The saturation function describes the minimum…

Given a graph $H$, we say that an edge-coloured graph $G$ is $H$-rainbow saturated if it does not contain a rainbow copy of $H$, but the addition of any non-edge in any colour creates a rainbow copy of $H$. The rainbow saturation number…

Let $F$ and $H$ be $k$-uniform hypergraphs. We say $H$ is $F$-saturated if $H$ does not contain a subgraph isomorphic to $F$, but $H+e$ does for any hyperedge $e\not\in E(H)$. The saturation number of $F$, denoted $\mathrm{sat}_k(n,F)$, is…

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$(2,q^4)$,…

In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following…

For given graphs $F$ and $G$, the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$ is called the $F$-saturation number and denoted $\mathrm{sat}(G, F)$. For the star $F=K_{1,r}$, the asymptotics of…