2 problems

Let $d\geq 1$, and let a measurable $k$-coloring of $\mathbb{R}^d$ mean a measurable partition into $k$ color classes. Two axis-aligned $d$-dimensional cubes are measure indiscerni…

Let $d,k,t\geq 1$, and let a measurable $k$-coloring of $\mathbb{R}^d$ assign one of $k$ colors to each point. A $d$-dimensional necklace is a measurable object in $\mathbb{R}^d$,…