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

For a family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$, let $\mathcal{D}(\mathcal{F}) = \{F\setminus G: F, G \in \mathcal{F}\}$ be the collection of all (setwise) differences of $\mathcal{F}$. The family $\mathcal{F}$ is called a…

For a family $\mathcal{F}$ of subsets of a finite set, define $\mathcal{D}(\mathcal{F})=\{F\setminus F': F, F'\in\mathcal{F}\}$. A family $\mathcal{F}$ is called intersecting if $F\cap F'\not=\emptyset$ for all $F, F'\in\mathcal{F}$. Frankl…

A family $\mathcal{A}$ of sets is said to be \emph{$t$-intersecting} if any two sets in $\mathcal{A}$ have at least $t$ common elements. A central problem in extremal set theory is to determine the size or structure of a largest…

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \emptyset$ for all $F, F' \in \mathcal F$. It is called almost…

We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the…

Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is…

Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is…

For a positive integer $d\geq 2$, a family $\mathcal F\subseteq \binom{[n]}{k}$ is said to be d-wise intersecting if $|F_1\cap F_2\cap \dots\cap F_d|\geq 1$ for all $F_1, F_2, \dots ,F_d\in \mathcal F$. A d-wise intersecting family…

A family $\mathcal{F}\subset \binom{[n]}{k}$ is called an intersecting family if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. If $\cap \mathcal{F}\neq \emptyset$ then $\mathcal{F}$ is called a star. The diversity of an…

A family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ is called a $t$-intersecting family if $|F\cap G| \geq t$ for any two members $F, G \in \mathcal{F}$ and for some positive integer $t$. If $t=1$, then we call the family $\mathcal{F}$…

Let $\mathcal{A}=\{A_{1},...,A_{p}\}$ and $\mathcal{B}=\{B_{1},...,B_{q}\}$ be two families of subsets of $[n]$ such that for every $i\in [p]$ and $j\in [q]$, $|A_{i}\cap B_{j}|= \frac{c}{d}|B_{j}|$, where $\frac{c}{d}\in [0,1]$ is an…

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 family $\mathcal{F}\subset \binom{[n]}{k}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The covering number of a family $\mathcal{F}$ is defined as the minimum size of $T\subset [n]$ such that $T\cap…

A family of $k$-subsets $A_1, A_2, ..., A_d$ on $[n]=\{1,2,..., n\}$ is called a $(d, c)$-cluster if the union $A_1\cup A_2 \cup ... \cup A_d$ contains at most $ck$ elements with $c<d$. Let $\mathcal{F}$ be a family of $k$-subsets of an…

The work in this article is concerned with two different types of families of finite sets: separating families and splitting families (they are also called "systems"). These families have applications in combinatorial search, coding theory,…

A family $\mathcal{F}$ of subsets of $[n]=\{1,2,\ldots,n\}$ shatters a set $A \subseteq [n]$ if for every $A' \subseteq A$ there is an $F \in \mathcal{F}$ such that $F \cap A=A'$. We develop a framework to analyze $f(n,k,d)$, the maximum…

Let $\mathcal{A}\subseteq{[n]\choose a}$ and $\mathcal{B}\subseteq{[n]\choose b}$ be two families of subsets of $[n]$, we say $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\neq \emptyset$ for all $A\in\mathcal{A}$,…

Diestel, et al. (see Order 35 (2017), JCT-A 167 (2019), arXiv:1805.01439) introduced the notion of abstract separation systems that satisfy a submodularity property, and they call this structural submodularity. Williamson, Goemans, Mihail,…

A $k$-wise $\ell$-divisible set family is a collection $\mathcal{F}$ of subsets of ${ \{1,\ldots,n \} }$ such that any intersection of $k$ sets in $\mathcal{F}$ has cardinality divisible by $\ell$. If $k=\ell=2$, it is well-known that…

Let $S$ be a family of sequences of positive numbers that decrease to 0, let $X$ be a metric space and $A \subset X$. $A$ is said to be $S$-dominated if, for every $s\in S$, a countable cover $\{E_n\}$ of $E$ can be found such that $diam…