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Register a new userA note on transitive union-closed families
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We show that the Union-Closed Conjecture holds for the union-closed family generated by the cyclic translates of any fixed set.
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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 $n$.
In this paper new infinite families of linear binary completely transitive codes are presented. They have covering radius $rho = 3$ and 4, and are a half part of the binary Hamming and the binary extended Hamming code of length $n=2^m-1$ and $2^m$, respectively, where $m$ is even. From these new completely transitive codes, in the usual way, i.e., as coset graphs, new presentations of infinite families of distance transitive coset graphs of diameter three and four, respectively, are constructed.
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 $h>0$.
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 most $s$. 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) $n=r+1$.
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 sets 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 $c$.
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