# On a Conjecture of Roth and Some Related Problems I

• P. Erdős
• A. Sárközy
• V. T. Sós
Chapter
Part of the Algorithms and Combinatorics 8 book series (AC, volume 8)

## 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)

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## Authors and Affiliations

• P. Erdős
• 1
• A. Sárközy
• 1
• V. T. Sós
• 1
1. 1.Math. Inst. of the Hung. Acad. of Sci.BudapestHungary