New bounds on the maximum number of neighborly boxes in Rd

Noga Alon, Jarosław Grytczuk, Andrzej P. Kisielewicz, Krzysztof Przesławski

Research output: Contribution to journalArticlepeer-review

Abstract

A family of axis-aligned boxes in Rd is 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)⩽(2−δ)d if k⩽(1−ɛ)d, where δ>0 depends on arbitrarily chosen ɛ>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 [Formula presented]. We also study k-neighborly families of boxes with additional structural properties. Families called 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.

Original languageEnglish
Article number103797
JournalEuropean Journal of Combinatorics
Volume114
DOIs
StatePublished - Dec 2023

Funding

FundersFunder number
National Science Foundation
Bloom's Syndrome Foundation2018267
Narodowe Centrum Nauki2020/37/B/ST1/03298

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