3 problems

Let $A\subset\mathbb{R}_{>0}$ be algebraically independent, meaning that no nonzero polynomial with rational coefficients vanishes on any finite tuple of distinct elements of $A$.…

Let a finite colouring of the plane be a colouring of $\mathbb{R}^2$ with finitely many colours, and say that it avoids a set of forbidden distances if no monochromatic pair of poi…

Let $D$ be a finite set of positive integers. A set $S\subseteq\mathds{N}$ is $D$-avoiding if no two elements of $S$ differ by an element of $D$; the germ-ordering compares sets th…