For $k,l$ positive integers let $h(k,l)$ be the least integer with the property that in a graph on $h(k,l)$ vertices either there is a closed circuit of $k$ or fewer lines, or the graph contains $l$ independent points.

It is given that for sufficiently large $l$, that $h(k,l)>l^{1+1/2k}$.

Now how do we conclude from here that for all $r$, there is an $r$-chromatic graph with no $k$-polygon in it?

Clearly a graph on $\lfloor r^{1+1/2k}\rfloor$ vertices will not have a $k$-polygon in it, but how can we construct it in such a way that it is also $r$-chromatic?