Related papers: Bisecting and D-secting families for set systems

For positive integers $n$ and $r$ such that $r \leq \lfloor n/2\rfloor$, let $X$ be a set of $n$ elements and let $\binom{X}{r}$ be the family of all $r$-subsets of $X$. Two sub-families $\mathcal{A}$ and $\mathcal{B}$ of $\binom{X}{r}$ are…

For a graph property $\mathcal{P}$ and a common vertex set $V = \{1, 2, \ldots, n\}$, a family of graphs on $V$ is \emph{$\mathcal{P}$-intersecting} iff $G \cap H$ satisfies $\mathcal{P}$ for all $G,H$ in the family. Addressing a question…

A (simple) hypergraph is a family H of pairwise incomparable sets of a finite set. We say that a hypergraph H is a domination hypergraph if there is at least a graph G such that the collection of minimal dominating sets of G is equal to H.…

Let $G=(V, E)$ be a simple and undirected graph. For some integer $k\geq 1$, a set $D\subseteq V$ is said to be a k-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $k$ neighbors in $D$. Furthermore, for some real…

A $k$-crossing family in a point set $S$ in general position is a set of $k$ segments spanned by points of $S$ such that all $k$ segments mutually cross. In this short note we present two statements on crossing families which are based on…

Let $B_\theta $ be the family of rectangles in the plane $R^2$, having slope $\theta $ with the abscissa. We say a set of slopes $\Theta $ is $D$-set if there exists a function $f\in L(R^2)$, such that the basis $B_\theta $ differentiates…

Let F be a finite nonempty family of finite nonempty sets. We prove the following: (i) F satisfies the condition of the title if and only if for every pair of distinct subfamilies {A_1,...,A_r}, {B_1,...,B_s} of F, the union of the A_i is…

A family $F$ of graphs on a fixed set of $n$ vertices is called triangle-intersecting if for any $G_1,G_2 \in F$, the intersection $G_1 \cap G_2$ contains a triangle. More generally, for a fixed graph $H$, a family $F$ is $H$-intersecting…

For $k\geq3$, a collection of $k$ sets is said to form a \emph{weak $\Delta$-system} if the intersection of any two sets from the collection has the same size. Erd\H{o}s and Szemer\'{e}di asked about the size of the largest family…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$.The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

The Galvin problem asks for the minimum size of a family $\mathcal{F} \subseteq \binom{[n]}{n/2}$ with the property that, for any set $A$ of size $\frac n 2$, there is a set $S \in \mathcal{F}$ which is balanced on $A$, meaning that $|S…

For a set $L$ of positive proper fractions and a positive integer $r \geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F} \subset \mathcal{P}([n])$ with the property that for any $2 \leq t \leq r$ and $A_1,…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

The Frankl's conjecture, formulated in 1979. and still open, states that in every family of sets closed for unions there is an element contained in at least half of the sets. A family Fc is called Frankl-complete (or FC-family) if in every…

The collection of branches (maximal linearly ordered sets of nodes) of the tree ${}^{<\omega}\omega$ (ordered by inclusion) forms an almost disjoint family (of sets of nodes). This family is not maximal -- for example, any level of the tree…

We say that a set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-$t$-intersecting if each set in $\mathcal{A}$ $t$-intersects each set…

We asymptotically determine the size of the largest family F of subsets of {1,...,n} not containing a given poset P if the Hasse diagram of P is a tree. This is a qualitative generalization of several known results including Sperner's…

Let $D$ be a simple digraph (directed graph) with vertex set $V(D)$ and arc set $A(D)$ where $n=|V(D)|$, and each arc is an ordered pair of distinct vertices. If $(v,u) \in A(D)$, then $u$ is considered an \emph{out-neighbor} of $v$ in $D$.…

Let $D$ be a strongly connected digraph. An arc set $S$ of $D$ is a \emph{restricted arc-cut} of $D$ if $D-S$ has a non-trivial strong component $D_{1}$ such that $D-V(D_{1})$ contains an arc. The \emph{restricted arc-connectivity}…

We prove that any definable family of subsets of a definable infinite set $A$ in an o-minimal structure has cardinality at most $|A|$. We derive some consequences in terms of counting definable types and existence of definable topological…