@Sawarnik Let $S$ be a set with $n$ elements.

Let $P(S)=\{X\mid X\subseteq S\}$

Let $H\subseteq\mathcal{P}(S)$ (hypergraph with edge set $S$).

Let $H_{|U}=\{U\cap A\mid A\in H\}$

Let $\dim_{\text{VC}}(H)=1+\max\{|U|\mid U\subseteq S\text{ and }H_{|U}=\mathcal{P}(U)\}$

Let $H'=H_{|S\backslash\{x\}}$

Let $H''=\{\alpha\in H'\mid\alpha\in H\text{ and }\alpha\cup\{x\}\in H\}$


Show that $\dim_{\text{VC}}(H')\leq\dim_{\text{VC}}(H)$ and $\dim_{\text{VC}}(H'')<\dim_{\text{VC}}(H)$

This geometry game I created is now moved to a new adress: http://www.euclidthegame.org
For the people interested.

And if people want to contribute by writing new levels, it is now on github:
https://github.com/kasperpeulen/euclidthegame

the only thin you need to add the javascript is something like:
checksegment("nameofyourtarget","nameofyourmessage")