A Problem of Erd\"{o}s Concerning Lattice Cubes

TL;DR

This paper investigates Erd"os's lattice cube problem, exploring maximum vertex sets avoiding rectangular corners, and introduces a two-dimensional matrix problem to analyze patterns and asymptotic behaviors related to the original conjecture.

Contribution

It provides initial computational results for small cubes, reformulates the problem in two dimensions, and uncovers formulas and patterns that may inform the original conjecture.

Findings

No unique maximal vertex set for small N

Reformulation to matrix problem with entries 0 and 1

Discovered formulas and asymptotic patterns

Abstract

This paper studies a problem of Erd\"{o}s concerning lattice cubes. Given an $N \times N \times N$ lattice cube, we want to find the maximum number of vertices one can select so that no eight corners of a rectangular box are chosen simultaneously. Erd\"{o}s conjectured that it has a sharp upper bound, which is $O(N^{11/4})$, but no example that large has been found yet. We start approaching this question for small $N$ using the method of exhaustion, and we find that there is not necessarily a unique maximal set of vertices (counting all possible symmetries). Next, we study an equivalent two-dimensional version of this problem looking for patterns that might be useful for generalizing to the three-dimensional case. Since an $n \times n$ grid is also an $n \times n$ matrix, we rephrase and generalize the original question to: what is the minimum number $\alpha(k,n)$ of vertices one can put in an $n \times n$ matrix with entries 0 and 1, such that every $k \times k$ minor contains at least one entry of 1, for $1 \leq k \leq n$? We discover some interesting formulas and asymptotic patterns that shed new light on the question.