ترغب بنشر مسار تعليمي؟ اضغط هنا

Note: Union-closed families with small average overlap densities

114   0   0.0 ( 0 )
 نشر من قبل David Ellis
 تاريخ النشر 2020
  مجال البحث
والبحث باللغة English
 تأليف David Ellis




اسأل ChatGPT حول البحث

In this very short note, we point out that the average overlap density of a union-closed family $mathcal{F}$ of subsets of ${1,2,ldots,n}$ may be as small as $Theta((log log |mathcal{F}|)/(log |mathcal{F}|))$, for infinitely many positive integers $n$.

قيم البحث

اقرأ أيضاً

We show that the Union-Closed Conjecture holds for the union-closed family generated by the cyclic translates of any fixed set.
Let $Asubset mathbb{N}^{n}$ be an $r$-wise $s$-union family, that is, a family of sequences with $n$ components of non-negative integers such that for any $r$ sequences in $A$ the total sum of the maximum of each component in those sequences is at mo st $s$. We determine the maximum size of $A$ and its unique extremal configuration provided (i) $n$ is sufficiently large for fixed $r$ and $s$, or (ii) $n=r+1$.
61 - Tom Eccles 2013
A family of sets is called union-closed if whenever $A$ and $B$ are sets of the family, so is $Acup B$. The long-standing union-closed conjecture states that if a family of subsets of $[n]$ is union-closed, some element appears in at least half the s ets of the family. A natural weakening is that the union-closed conjecture holds for large families; that is, families consisting of at least $p_02^n$ sets for some constant $p_0$. The first result in this direction appears in a recent paper of Balla, Bollobas and Eccles cite{BaBoEc}, who showed that union-closed families of at least $frac{2}{3}2^n$ sets satisfy the conjecture --- they proved this by determining the minimum possible average size of a set in a union-closed family of given size. However, the methods used in that paper cannot prove a better constant than $frac{2}{3}$. Here, we provide a stability result for the main theorem of cite{BaBoEc}, and as a consequence we prove the union-closed conjecture for families of at least $(frac{2}{3}-c)2^n$ sets, for a positive constant $c$.
120 - Chuanqi Xiao , Oscar Zamora 2020
The Turan number of a graph $H$, $text{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. A wheel $W_n$ is an $n$-vertex graph formed by connecting a single vertex to all vertices of a cycle $C _{n-1}$. Let $mW_{2k+1}$ denote the $m$ vertex-disjoint copies of $W_{2k+1}$. For sufficiently large $n$, we determine the Turan number and all extremal graphs for $mW_{2k+1}$. We also provide the Turan number and all extremal graphs for $W^{h}:=bigcuplimits^m_{i=1}W_{k_i}$ when $n$ is sufficiently large, where the number of even wheels is $h$ and $h>0$.
72 - Mohit Kumbhat , Kevin Moss , 2015
List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection of adjacent lists is sufficiently small. We introduce a new refinement, called choosability with union separation, where we require that the union of adjacent lists is sufficiently large. For $t geq k$, a $(k,t)$-list assignment is a list assignment $L$ where $|L(v)| geq k$ for all vertices $v$ and $|L(u)cup L(v)| geq t$ for all edges $uv$. A graph is $(k,t)$-choosable if there is a proper coloring for every $(k,t)$-list assignment. We explore this concept through examples of graphs that are not $(k,t)$-choosable, demonstrating sparsity conditions that imply a graph is $(k,t)$-choosable, and proving that all planar graphs are $(3,11)$-choosable and $(4,9)$-choosable.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا