A new upper bound on the number of neighborly boxes in R^d

A new upper bound on the number of neighborly boxes in R^d is given. We apply a classical result of Kleitman on the maximum size of sets with a given diameter in discrete hypercubes. We also present results of some computational experiments and an emerging conjecture

A note on a flip-connected class of generalized domino tilings of the box [0,2]n[0,2]^n

Let $n,d\in \mathbb{N}$ and $n>d$. An $(n-d)$-domino is a box $I_1\times \cdots \times I_n$ such that $I_j\in \{[0,1],[1,2]\}$ for all $j\in N\subset [n]$ with $|N|=d$ and $I_i=[0,2]$ for every $i\in [n]\setminus N$. If $A$ and $B$ are two $(n-d)$-dominoes such that $A\cup B$ is an $(n-(d-1))$-domino, then $A,B$ is called a twin pair. If $C,D$ are two $(n-d)$-dominoes which form a twin pair such that $A\cup B=C\cup D$ and $\{C,D\}\neq \{A,B\}$, then the pair $C,D$ is called a flip of $A,B$. A family $\mathscr{D}$ of $(n-d)$-dominoes is a tiling of the box $[0,2]^n$ if interiors of every two members of $\mathscr{D}$ are disjoint and $\bigcup_{B\in \mathscr{D}}B=[0,2]^n$. An $(n-d)$-domino tiling $\mathscr{D}'$ is obtained from an $(n-d)$-domino tiling $\mathscr{D}$ by a flip, if there is a twin pair $A,B\in \mathscr{D}$ such that $\mathscr{D}'=(\mathscr{D}\setminus \{A,B\})\cup \{C,D\}$, where $C,D$ is a flip of $A,B$. A family of $(n-d)$-domino tilings of the box $[0,2]^n$ is flip-connected, if for every two members $\mathscr{D},\mathscr{E}$ of this family the tiling $\mathscr{E}$ can be obtained from $\mathscr{D}$ by a sequence of flips. In the paper some flip-connected class of $(n-d)$-domino tilings of the box $[0,2]^n$ is described

Partition problems in high dimensional boxes

Alon, Bohman, Holzman and Kleitman proved that any partition of a $d$-dimensional discrete box into proper sub-boxes must consist of at least $2^d$ sub-boxes. Recently, Leader, Mili\'{c}evi\'{c} and Tan considered the question of how many odd-sized proper boxes are needed to partition a $d$-dimensional box of odd size, and they asked whether the trivial construction consisting of $3^d$ boxes is best possible. We show that approximately $2.93^d$ boxes are enough, and consider some natural generalisations.Comment: 19 pages, 10 figure