Because I'm not used to the theorem of VC-dimension, I don't know why these statements hold.
Why axis parallel square can shatter a set of three points?
and can't shatter any set of four points?
2026-03-30 17:06:31.1774890391
Question about VC-dimension
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If you have $3$ points that form triangle, it should be clear that you can enclose every point with an square, you just have to make it as small as you need and it is also clear that there must happen that $x_1\leq x_2<x_3$ with either $x_1<x_2$ or $x_2<x_3$(because they are collinear) so you can separate two points.
The problem with $4$ points is that if they are collinear, and you order them somehow in that line say by $x_1<x_2<x_3<x_4,$ then is not possible to have $x_1,x_3$ separated from $x_2,x_4.$ Because $x_2$ has to lie in any square containing $x_1,x_3.$