Prove a subset from 1000 points contains one point that is strictly larger than the other one
For those curious about why the number of equivalence classes is 271 = $10^3$ - $9^3$, consider which elements of $[10]^3$ are NOT representatives of their equivalence class (i.e. their least coordinate is not a 1). It is all the points with all three coordinates of in the range of 2 to 9, of which there are $9^3$.