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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 448.10002

**Autor: ** Erdös, Paul

**Title: ** A survey of problems in combinatorial number theory. (In English)

**Source: ** Ann. Discrete Math. 6, 89-115 (1980).

**Review: ** Problems are presented in the following areas: the theorem of van der Waerden and Szemerédi; covering congruences, additive number theory, dense sets of integers, infinite subsets of integers (related to the work of Hindmann), sieve methods and other miscellaneous topics. An attempt is made to describe what has happened to problems mentioned in previous surveys. Some old conjectures still remain open, for example the following one of 45 years standing: if 1 \leq a_{1} < ... < a_{k} \leq x is a sequence of integers such that the sums **sum**_{1}^{k}\epsilon_{i}a_{i}, \epsilon_{i} = 0or1, are all different, then **max** k = \frac{log x}{log 2}+0(1). 500 dollars is coffered for its solution. A larger survey due to the author and R. L. Graham can be found in ''Old and new problems and results in combinatorial number theory'' [L'Enseignment Math., Monographie 28 (1980; Zbl 434.10001)].

**Reviewer: ** I.Anderson

**Classif.: ** * 11-02 Research monographs (number theory)

11B25 Arithmetic progressions

11B83 Special sequences of integers and polynomials

11P32 Additive questions involving primes

11N05 Distribution of primes

11A99 Elementary number theory

00A07 Problem books

05A05 Combinatorial choice problems

**Keywords: ** problems; covering congruences; dense sets of integers; infinite subsets of integers; sieve methods

**Citations: ** Zbl.434.10001

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag