Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMultiply union families in $mathbb{N}^n$
176
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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) $n=r+1$.
rate research
Read More
A subspace of $mathbb{F}_2^n$ is called cyclically covering if every vector in $mathbb{F}_2^n$ has a cyclic shift which is inside the subspace. Let $h_2(n)$ denote the largest possible codimension of a cyclically covering subspace of $mathbb{F}_2^n$. We show that $h_2(p)= 2$ for every prime $p$ such that 2 is a primitive root modulo $p$, which, assuming Artins conjecture, answers a question of Peter Cameron from 1991. We also prove various bounds on $h_2(ab)$ depending on $h_2(a)$ and $h_2(b)$ and extend some of our results to a more general set-up proposed by Cameron, Ellis and Raynaud.
We show that the Union-Closed Conjecture holds for the union-closed family generated by the cyclic translates of any fixed set.
Hefetz, M{u}tze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper we support the analogous question for distance magic labeling. Let $Gamma$ be an Abelian group of order $n$. A textit{directed $Gamma$-distance magic labeling} of an oriented graph $vec{G} = (V,A)$ of order $n$ is a bijection $vec{l}:V rightarrow Gamma$ with the property that there is a textit{magic constant} $mu in Gamma$ such that for every $x in V(G)$ $ w(x) = sum_{y in N^{+}(x)}vec{l}(y) - sum_{y in N^{-}(x)} vec{l}(y) = mu. $ In this paper we provide an infinite family of odd regular graphs possessing an orientable $mathbb{Z}_{n}$-distance magic labeling. Our results refer to lexicographic product of graphs. We also present a family of odd regular graphs that are not orientable $mathbb{Z}_{n}$-distance magic.
For each natural number $d$, we introduce the concept of a $d$-cap in $mathbb{F}_3^n$. A subset of $mathbb{F}_3^n$ is called a $d$-cap if, for each $k = 1, 2, dots, d$, no $k+2$ of the points lie on a $k$-dimensional flat. This generalizes the notion of a cap in $mathbb{F}_3^n$. We prove that the $2$-caps in $mathbb{F}_3^n$ are exactly the Sidon sets in $mathbb{F}_3^n$ and study the problem of determining the size of the largest $2$-cap in $mathbb{F}_3^n$.
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 $n$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
