New bounds on the maximum number of neighborly boxes in R^d
Abstract
A family of axis-aligned boxes in $\er^d$ is \emph{$k$-neighborly} if the intersection of every two of them has dimension at least $d-k$ and at most $d-1$. Let $n(k,d)$ denote the maximum size of such a family. It is known that $n(k,d)$ can be equivalently defined as the maximum number of vertices in a complete graph whose edges can be covered by $d$ complete bipartite graphs, with each edge covered at most $k$ times. We derive a new upper bound on $n(k,d)$, which implies, in particular, that $n(k,d)\leqslant (2-\delta)^d$ if $k\leqslant (1-\varepsilon)d$, where $\delta>0$ depends on arbitrarily chosen $\varepsilon>0$. The proof applies a classical result of Kleitman, concerning the maximum size of sets with a given diameter in discrete hypercubes. By an explicit construction we obtain also a new lower bound for $n(k,d)$, which implies that $n(k,d)\geqslant (1-o(1))\frac{d^k}{k!}$. We also study $k$-neighborly families of boxes with additional structural properties. Families called \emph{total laminations}, that split in a tree-like fashion, turn out to be particularly useful for explicit constructions. We pose a few conjectures based on these constructions and some computational experiments.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2022
- DOI:
- 10.48550/arXiv.2212.05133
- arXiv:
- arXiv:2212.05133
- Bibcode:
- 2022arXiv221205133A
- Keywords:
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- Mathematics - Combinatorics