# Maximal sets of given diameter in the grid and the torus

## Abstract

The grid graph is the graph on [k]n={0,1,...k-1}n in which x=(x1,...,xn) is joined to y=(y1,...,yn) if for some j we have |xj-yj|=1 and xi=yi for all i≠j. One of our aims in this paper is to determine, for each positive integer d, the maximum size of a subset of [k]n of diameter d. The discrete torus is the corresponding graph on Znk=( Z kZ)n: a point x=(xi)n1 is joined to y=(yi)n1 if for some j we have xj=yj±1 and xi=yi for all i≠ j. Our other main aim is to determine, for each d, the maximum size of a subset of Znk of diameter d, in the case k even. © 1993.

## Publication Title

Discrete Mathematics

## Recommended Citation

Bollobás, B., & Leader, I.
(1993). Maximal sets of given diameter in the grid and the torus.* Discrete Mathematics**, 122* (1-3), 15-35.
https://doi.org/10.1016/0012-365X(93)90284-Z