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n\sqrt{n} lattice points

Source: KoMaL A.823

April 12, 2022
combinatoricslattice points

Problem Statement

For positive integers nn consider the lattice points Sn={(x,y,z):1≤x≤n,1≤y≤n,1≤z≤n,x,y,z∈N}.S_n=\{(x,y,z):1\le x\le n, 1\le y\le n, 1\le z\le n, x,y,z\in \mathbb N\}. Is it possible to find a positive integer nn for which it is possible to choose more than nnn\sqrt{n} lattice points from SnS_n such that for any two chosen lattice points at least two of the coordinates of one is strictly greater than the corresponding coordinates of the other?
[I]Proposed by Endre Csóka, Budapest
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