Related papers: Erd\"os-Ko-Rado sets of flags of finite sets

A flag of a finite set $S$ is a set $f$ of non-empty, proper subsets of $S$, such that $X\subseteq Y$ or $Y\subseteq X$ for all $X,Y\in f$. Two flags $f_1$ and $f_2$ of $S$ are opposite if $X_1\cap X_2=\emptyset$, or $X_1\cup X_2=S$ for all…

Let $S$ be a non-empty finite set. A flag of $S$ is a set $f$ of non-empty proper subsets of $S$ such that $X\subseteq Y$ or $Y\subseteq X$ for all $X,Y\in f$. The set $\{|X|:X\in f\}$ is called the type of $f$. Two flags $f$ and $f'$ are…

We continue our investigation of Erd\H{o}s-Ko-Rado (EKR) sets of flags in spherical buildings. In previous work, we used the theory of buildings and Iwahori-Hecke algebras to obtain upper bounds on their size. As the next step towards the…

In this paper we study a question related to the classical Erd\H{o}s-Ko-Rado theorem, which states that any family of $k$-element subsets of the set $[n] = \{1,\ldots,n\}$ in which any two sets intersect, has cardinality at most…

The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the…

We prove an analogue of the classical Erd\H{o}s-Ko-Rado theorem for intersecting sets of permutations in finite 2-transitive groups. Given a finite group G acting faithfully and 2-transitively on the set X, we show that an intersecting set…

The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $F\subseteq V(G)$ such that only the trivial automorphism of $G$ fixes every vertex in $F$. Let $\Pi$ $=$ $\{F_1,F_2,\ldots,F_k\}$ be an ordered $k$-partition…

A fair dominating set in a graph $G$ (or FD-set) is a dominating set $S$ such that all vertices not in $S$ are dominated by the same number of vertices from $S$; that is, every two vertices not in $S$ have the same number of neighbors in…

In this note, we investigate some properties of local Kneser graphs defined in [8]. In this regard, as a generalization of the Erd${\rm \ddot{o}}$s-Ko-Rado theorem, we characterize the maximum independent sets of local Kneser graphs. Next,…

A chamber of the vector space $\mathbb{F}_q^n$ is a set $\{S_1,\dots,S_{n-1}\}$ of subspaces of $\mathbb{F}_q^n$ where $S_1\subset S_2\subset \dotso \subset S_{n-1}$ and $\dim(S_i)=i$ for $i=1,\dots,n-1$. By $\Gamma_n(q)$ we denote the…

The vertex set of the Kneser graph $K(n,k)$ is $V = \binom{[n]}{k}$ and two vertices are adjacent if the corresponding sets are disjoint. For any graph $F$, the largest size of a vertex set $U \subseteq V$ such that $K(n,k)[U]$ is $F$-free,…

Consider classical Kneser's graph $K(n,r)$: for two natural numbers $ r, n $ such that $r \le n / 2$, its vertices are all the subsets of $[n]=\{1,2,\ldots,n\}$ of size $r$, and two such vertices are adjacent if the corresponding subsets…

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called…

Getting inspired by the famous no-three-in-line problem and by the general position subset selection problem from discrete geometry, the same is introduced into graph theory as follows. A set $S$ of vertices in a graph $G$ is a general…

In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erd\H{o}s-Ko-Rado theorem for…

A family of subsets $\mathcal{F}\subseteq {[n]\choose k}$ is called intersecting if any two of its members share a common element. Consider an intersecting family, a direct problem is to determine its maximal size and the inverse problem is…

A family of subsets $\mathcal{F}$ is intersecting if $A \cap B \neq \emptyset$ for any $A, B \in \mathcal{F}$. In this paper, we show that for given integers $k > d \ge 2$ and $n \ge 2k+2d-3$, and any intersecting family $\mathcal{F}$ of…

A dominating set of a graph $G$ is a set of vertices $D$ such that for all $v \in V(G)$, either $v \in D$ or $(v,d) \in E(G)$ for some $d \in D$. The cardinality redundance of a vertex set $S$, $CR(S)$, is the number of vertices in $V(G)$…

Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in $G=\left( V,E\right) $. Let $\xi(G)$ denote the size of the intersection of all maximum independent sets. It is known…

The general position problem is to find the cardinality of a largest vertex subset S such that no triple of vertices of S lie on a common geodesic. For a connected graph G, the cardinality of S is denoted by gp(G) and called gp-number (or…