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On a Conjecture of Roth and Some Related Problems I

  • P. Erdős
  • A. Sárközy
  • V. T. Sós
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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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

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

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