RO-TST 2015 Problem 9

Consider the integral lattice $\mathbb{Z}^n$ , $n \geq 2$ , in the Euclidean $n$ -space. Define a line in $\mathbb{Z}^n$ to be a set of the form $a_1 \times \cdots \times a_{k-1} \times \mathbb{Z} \times a_{k+1} \times \cdots \times a_n$ where $k$ is an integer in the range $1,2,\ldots,n$ , and the $a_i$ are arbitrary integers. A subset $A$ of $\mathbb{Z}^n$ is called admissible if it is non-empty, finite, and every line in $\mathbb{Z}^{n}$ which intersects $A$ contains at least two points from $A$ . A subset $N$ of $\mathbb{Z}^n$ is called null if it is non-empty, and every line in $\mathbb{Z}^n$ intersects $N$ in an even number of points (possibly zero).(a) Prove that every admissible set in $\mathbb{Z}^2$ contains a null set.(b) Exhibit an admissible set in $\mathbb{Z}^3$ no subset of which is a null set .