6 problems

Let $A$ be a sequence of integers. We call $A$ an $R$-sequence for totally minimal systems if, for every totally minimal system $(X,T)$, there exists $x\in X$ such that the orbit o…

Glasner–Huang–Shao–Weiss–Ye conjecture. The set $R$ has nonempty intersection with every infinite arithmetic progression of step size $k$.

Let $G$ be an amenable group, ${G \curvearrowright X}$ a minimal action, and let $k\geq 2$. Write ${\rm IT}_k(X,G)$ for the set of independence tuples of length $k$, let…

Let $(X,T)$ be a minimal PI topological dynamical system, and let $x\in X$. The point $(x,x)$ is minimal for the system $(X\times X,T\times T^2)$ if and only if, for every neighbor…

Pro-nilfactor conjecture. If $(X,T)$ is a distal minimal system, then its maximal $(d-1)$-step pro-nilfactor is its topological characteristic factor along arithmetic progressions…

A universal symbolic system is a symbolic dynamical system capable of simulating every effective symbolic system. Universality conjecture. Every universal symbolic system has infin…