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Register a new userCyclically covering subspaces in $mathbb{F}_2^n$
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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.
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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 $mathbb{F}_3^n$.
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) $n=r+1$.
We first show that the subgroup of the abelian real group $mathbb{R}$ generated by the coordinates of a point in $x = (x_1,dots,x_n)inmathbb{R}^n$ completely classifies the $mathsf{GL}(n,mathbb Z)$-orbit of $x$. This yields a short proof of J.S.Danis theorem: the $mathsf{GL}(n,mathbb Z)$-orbit of $xinmathbb{R}^n$ is dense iff $x_i/x_jin mathbb{R} setminus mathbb Q$ for some $i,j=1,dots,n$. We then classify $mathsf{GL}(n,mathbb Z)$-orbits of rational affine subspaces $F$ of $mathbb{R}^n$. We prove that the dimension of $F$ together with the volume of a special parallelotope associated to $F$ yields a complete classifier of the $mathsf{GL}(n,mathbb Z)$-orbit of $F$.
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.
We address a long-standing and long-investigated problem in combinatorial topology, and break the exponential barrier for triangulations of real projective space, constructing a trianglation of $mathbb{RP}^n$ of size $e^{(frac{1}{2}+o(1))sqrt{n}{log n}}$.
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