5 problems

Ergodic variant of the quadratic Rado conjecture. There exist distinct $x,y,z\in\mathbb{N}$ such that

Quadratic Rado conjecture. The equation

Generalized Pythagorean-pair conjecture. If at least one of $ac$, $bc$, or $(a+b)c$ is a square, then $ax^2+by^2=cz^2$ is partition regular with respect to $x,y$.

Rado triple conjecture. The equation

FKM's pairwise partition-regularity conjecture. If at least one of $ac$, $bc$, or $(a+b)c$ is a square, then $a x^2+b y^2=c z^2$ is partition regular with respect to $x,y$.