On a Conjecture of Roth and Some Related Problems I

Abstract

Let N denote the set of positive integers and put ${\text{[1,}}{\rm N}{\text{] = \{ 1,}}...{\text{,}}{\rm N}{\text{\} }}$. We use |S| to denote the cardinality of the finite set S. If S is a given set and $\mathcal A_1,...,\mathcal A_k$ are subsets of S with

$$S={\cup}_{i=1}^k\mathcal{A}_i,\ \mathcal{A}_i\cap\mathcal{A}_i=\Theta for\ \i\neq j,$$

then $\{\mathcal A_1,...,\mathcal A_k\}$ will be called a k-partition (or k-colouring) of S, and the subsets $\mathcal A_1,...,\mathcal A_k$ will be referred to as classes. Let $f:\mathcal N^t\rightarrow \mathcal N$ be a given function. If

$$n = f(a_1,...,a_t)$$

with $a_1,..., a_t$ belonging to the same class, then this will be called a monochromatic representation of n in the form (1)

Research partially supported by Hungarian National Foundation for Scientific Research grant no. 1811