A note on three dimensional good sets

TL;DR

This paper investigates the structure of good sets in three-dimensional Cartesian products, revealing that certain components may not be full components, similar to higher dimensions.

Contribution

It extends the understanding of good sets from higher dimensions to the three-dimensional case, showing a key structural difference.

Findings

In 3-fold Cartesian products, related components of good sets need not be full components.

The behavior of good sets in three dimensions parallels that in higher dimensions for certain properties.

Abstract

We show that as in the case of n- fold Cartesian product for n greater than or equal to 4, even in 3-fold Cartesian product, a related component of a good set need not be a full component.