ﻻ يوجد ملخص باللغة العربية
Let $mathcal{F}$ and $mathcal{G}$ be two $t$-uniform families of subsets over $[k] = {1,2,...,k}$, where $|mathcal{F}| = |mathcal{G}|$, and let $C$ be the adjacency matrix of the bipartite graph whose vertices are the subsets in $mathcal{F}$ and $mathcal{G}$, and there is an edge between $Ain mathcal{F}$ and $B in mathcal{G}$ if and only if $A cap B eq emptyset$. The pair $(mathcal{F},mathcal{G})$ is $q$-almost cross intersecting if every row and column of $C$ has exactly $q$ zeros. We consider $q$-almost cross intersecting pairs that have a circulant intersection matrix $C_{p,q}$, determined by a column vector with $p > 0$ ones followed by $q > 0$ zeros. This family of matrices includes the identity matrix in one extreme, and the adjacency matrix of the bipartite crown graph in the other extreme. We give constructions of pairs $(mathcal{F},mathcal{G})$ whose intersection matrix is $C_{p,q}$, for a wide range of values of the parameters $p$ and $q$, and in some cases also prove matching upper bounds. Specifically, we prove results for the following values of the parameters: (1) $1 leq p leq 2t-1$ and $1 leq q leq k-2t+1$. (2) $2t leq p leq t^2$ and any $q> 0$, where $k geq p+q$. (3) $p$ that is exponential in $t$, for large enough $k$. Using the first result we show that if $k geq 4t-3$ then $C_{2t-1,k-2t+1}$ is a maximal isolation submatrix of size $ktimes k$ in the $0,1$-matrix $A_{k,t}$, whose rows and columns are labeled by all subsets of size $t$ of $[k]$, and there is a one in the entry on row $x$ and column $y$ if and only if subsets $x,y$ intersect.
A family $mathcal F$ has covering number $tau$ if the size of the smallest set intersecting all sets from $mathcal F$ is equal to $s$. Let $m(n,k,tau)$ stand for the size of the largest intersecting family $mathcal F$ of $k$-element subsets of ${1,ld
For a family $mathcal F$, let $mathcal D(mathcal F)$ stand for the family of all sets that can be expressed as $Fsetminus G$, where $F,Gin mathcal F$. A family $mathcal F$ is intersecting if any two sets from the family have non-empty intersection. I
A family of sets is said to be emph{symmetric} if its automorphism group is transitive, and emph{intersecting} if any two sets in the family have nonempty intersection. Our purpose here is to study the following question: for $n, kin mathbb{N}$ with
We shall be interested in the following Erdos-Ko-Rado-type question. Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic translate (modulo n) of B? Chung, Grah
A family of perfect matchings of $K_{2n}$ is $t$-$intersecting$ if any two members share $t$ or more edges. We prove for any $t in mathbb{N}$ that every $t$-intersecting family of perfect matchings has size no greater than $(2(n-t) - 1)!!$ for suffic