Erdős, Paul; Szemerédi, E. On sums and products of integers. (English) Zbl 0526.10011 Studies in Pure Mathematics, Mem. of P. Turán, 213-218 (1983). [This article was published in the book announced in Zbl 0512.00007.] Denoting by \(f(n)\) the largest integer such that for every \(\{1\leq a_1\leq\dots\leq a_n\}\) integer set there are at least \(f(n)\) distinct numbers of the form \(a_i+a_j\), \(a_ia_j\), \(1\leq i\leq j\leq n\), the authors prove that \[ n^{1+c_1}< f(n)< n^2\exp(-c_2\log n/\log\log n). \] Some other related results and a lot of related conjectures are also discussed. The proof is self-contained and based only on elementary combinatorial arguments. Reviewer: Antal Balog (Budapest) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 21 ReviewsCited in 101 Documents MSC: 11B75 Other combinatorial number theory 11B30 Arithmetic combinatorics; higher degree uniformity Keywords:sums and products of integers; combinatorial number theory; addition and multiplication of sets Citations:Zbl 0512.00007 × Cite Format Result Cite Review PDF Online Encyclopedia of Integer Sequences: Cardinality of the union of the set of sums and the set of products made from pairs of integers from {1..n}. Smallest possible cardinality of the union of the set of pairwise sums and the set of pairwise products from a set of n positive integers.