Extremal subgraphs of the $d$-dimensional grid graph

TL;DR

This paper determines the maximum number of edges in induced subgraphs of a $d$-dimensional grid graph for any size, providing both exact and asymptotic bounds, generalizing previous 2D and small-size results.

Contribution

It offers a comprehensive analysis of extremal subgraphs in $d$-dimensional grid graphs, extending known results to higher dimensions and all subgraph sizes.

Findings

Exact bounds for maximum edges in subgraphs of any size

Asymptotic bounds derived using Bollobás and Thomason's theorem

Generalization of previous 2D and small-size results

Abstract

For each natural number $n$ we determine, both asymptotically and exactly, the maximum number of edges an induced subgraph of order $n$ of the $d$-dimension a grid graph ${\ints}^d$ can have. The asymptotic bound is obtained by using a theorem Bollob\'{a}s and Thomason, and the exact bound is obtained by induction. This generalizes some earlier results for the case $d=2$ on one hand, and for $n\leq 2^d$ on the other.