2 problems

Let $N^n_m$ be the subset of $\mathbb{R}^n$ consisting of points with at most $m$ rational coordinates. Chigogidze's conjecture. Every $m$-dimensional subset of $\mathbb{R}^n$ can…

Let $S$ be a $p$-punctured surface of genus $g$, and set $n=3g+p-4$. Define $k$ by $k=0$ if $g=0$, $k=g-1$ if both $p\neq 0$ and $g\neq 0$, and $k=g-2$ if $p=0$. Let…