# Neighborly boxes and strings with jokers; constructions and asymptotics

## TL;DR

This paper investigates the maximum size of families of axis-aligned boxes in high-dimensional space with intersection dimension constraints, providing new constructions and asymptotic bounds especially when the intersection parameter is close to the dimension.

## Contribution

The authors introduce a novel construction of $k$-neighborly boxes that improves lower bounds for large $k$, and establish asymptotic formulas for the maximum number of such boxes.

## Key findings

- New construction improves lower bounds for $n(k,d)$ when $k$ is close to $d$.
- Asymptotic formula for $n(d-s,d)$ as $d$ grows large.
- Use of Hamming cube interpretation with joker symbols in constructions.

## Abstract

We study families of axis-aligned boxes in a $d$-dimensional Euclidean space $\mathbb{R}^d$ whose placement is restricted by bounds on the dimension of their pairwise intersections. More specifically, two such boxes in $\mathbb{R}^d$ are said to be \emph{$k$-neighborly} if their intersection has dimension at least $d-k$ and at most $d-1$. The maximum number of pairwise $k$-neighborly boxes in $\mathbb{R}^d$ is denoted by $n(k,d)$. It is known that $n(k,d)=\Theta(d^k)$, for fixed $1\leqslant k\leqslant d$, however, exact formulas are known only in three cases: $k=1$, $k=d-1$, and $k=d$. In particular, the equality $n(1,d)=d+1$ is equivalent to the famous theorem of Graham and Pollak concerning partitions of complete graphs into complete bipartite graphs. In our main result we give a new construction of families of $k$-neighborly boxes which improves the lower bound for $n(k,d)$ when $k$ is close to $d$. Together with some recent upper bounds on $n(k,d)$, it gives the asymptotic equality $n(d-s,d)\thicksim\frac{2^s+1}{2^{s+1}}\cdot2^d$, for every fixed $s\leqslant d/2$. In our constructions we use a familiar interpretation of the problem in the language of Hamming cubes represented by binary strings with a special blank symbol, called \emph{joker}.