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Admissible and null sets !!!

Source: Romania TST 2015 Day 2 Problem 4

June 4, 2015
combinatoricsRomanian TST

Problem Statement

Consider the integral lattice Zn\mathbb{Z}^n, n≥2n \geq 2, in the Euclidean nn-space. Define a line in Zn\mathbb{Z}^n to be a set of the form a1×⋯×ak−1×Z×ak+1×⋯×ana_1 \times \cdots \times a_{k-1} \times \mathbb{Z} \times a_{k+1} \times \cdots \times a_n where kk is an integer in the range 1,2,…,n1,2,\ldots,n, and the aia_i are arbitrary integers. A subset AA of Zn\mathbb{Z}^n is called admissible if it is non-empty, finite, and every line in Zn\mathbb{Z}^{n} which intersects AA contains at least two points from AA. A subset NN of Zn\mathbb{Z}^n is called null if it is non-empty, and every line in Zn\mathbb{Z}^n intersects NN in an even number of points (possibly zero). (a) Prove that every admissible set in Z2\mathbb{Z}^2 contains a null set. (b) Exhibit an admissible set in Z3\mathbb{Z}^3 no subset of which is a null set .
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