Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userUniform intersecting families with large covering number
77
0
0.0
(
0
)
Added by
Andrey Kupavskii
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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 ${1,ldots,n}$ with covering number $tau$. It is a classical result of ErdH os and Lovasz that $m(n,k,k)le k^k$ for any $n$. In this short note, we explore the behaviour of $m(n,k,tau)$ for $n<k^2$ and large $tau$. The results are quite surprising: For example, we show that $m(k^{3/2},k,tau) = (1-o(1)){n-1choose k-1}$ for $taule k-k^{3/4+o(1)}$. At the same time, $m(k^{3/2},k,tau)<e^{-ck^{1/2}}{nchoose k}$ if $tau>k-frac 12k^{1/2}$.
rate research
Read More
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. In this paper, we study the following question: what is the maximum of $|mathcal D(mathcal F)|$ for an intersecting family of $k$-element sets? Frankl conjectured that the maximum is attained when $mathcal F$ is the family of all sets containing a fixed element. We show that this holds if $n>50klog k$ and $k>50$. At the same time, we provide a counterexample for $n< 4k.$
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.
Let $r(k)$ denote the maximum number of edges in a $k$-uniform intersecting family with covering number $k$. ErdH{o}s and Lovasz proved that $ lfloor k! (e-1) rfloor leq r(k) leq k^k.$ Frankl, Ota, and Tokushige improved the lower bound to $r(k) geq left( k/2 right)^{k-1}$, and Tuza improved the upper bound to $r(k) leq (1-e^{-1}+o(1))k^k$. We establish that $ r(k) leq (1 + o(1)) k^{k-1}$.
For positive integers $n,r,k$ with $nge r$ and $kge2$, a set ${(x_1,y_1),(x_2,y_2),dots,(x_r,y_r)}$ is called a $k$-signed $r$-set on $[n]$ if $x_1,dots,x_r$ are distinct elements of $[n]$ and $y_1dots,y_rin[k]$. We say a $t$-intersecting family consisting of $k$-signed $r$-sets on $[n]$ is trivial if each member of this family contains a fixed $k$-signed $t$-set. In this paper, we determine the structure of large maximal non-trivial $t$-intersecting families. In particular, we characterize the non-trivial $t$-intersecting families with maximum size for $tge2$, extending a Hilton-Milner-type result for signed sets given by Borg.
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 $k le n/2$, how large can a symmetric intersecting family of $k$-element subsets of ${1,2,ldots,n}$ be? As a first step towards a complete answer, we prove that such a family has size at most [expleft(-frac{c(n-2k)log n}{k( log n - log k)} right) binom{n}{k},] where $c > 0$ is a universal constant. We also describe various combinatorial and algebraic approaches to constructing such families.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
