5 problems

Monotonicity conjecture. There exists an integer $s>r$ such that

Let $1\leq a\leq b$ be integers. An $(a,b)$-triple is a triple $(x,ax+d,bx+2d)$ with positive integers $x,d$. The pair $(a,b)$ is regular if, for every $r\in\mathbb{N}$, there is a…

Let $1\leq a\leq b$ be integers, and let $\operatorname{dor}(a,b)$ denote the degree of regularity of the family of $(a,b)$-triples $(x,ax+d,bx+2d)$ with positive integers $x,d$. T…

Let $1\leq a\leq b$ be integers. An $(a,b)$-triple is a triple of the form $(x,ax+d,bx+2d)$, where $x,d$ are positive integers. The degree of regularity of the family of all…

Let $r,k,g\tinge$ be positive integers. A set $A\subseteq \mathbb{N}$ is $(r,k)$-van der Waerden if every $r$-colouring of $A$ contains a monochromatic arithmetic progression of le…