Question 177
Given is an odd integer $n \ge 1$. Let $S$ be the set of all points in the three dimensional space, whose all coordinates belong to the set $\{0, 1, ..., n\}$. Determine the maximum size of a subset $A \subset S$ with the following property: For every two distinct points $(x_1, x_2, x_3)$, $(y_1, y_2, y_3) \in A$ among the three numbers $x_1 - y_1, x_2 - y_2, x_3 - y_3$ there is at least one positive number and at least one negative number.