Journal of Integer Sequences, Vol. 18 (2015), Article 15.7.7

Sets of Natural Numbers with Proscribed Subsets


Kevin O'Bryant
Department of Mathematics
College of Staten Island (CUNY)
Staten Island, NY 10314
USA

Abstract:

Let 𝒜 be a set of subsets of the natural numbers, and let G𝒜(n) be the maximum cardinality of a subset of {1, 2, . . . , n} that does not have any subsets that are in 𝒜. We consider the general problem of giving upper bounds on G𝒜(n), and give new results for some 𝒜 that are closed under dilation. We specifically address some examples, including sets that do not contain geometric progressions of length k with integer ratio, sets that do not contain geometric progressions of length k with rational ratio, and sets of integers that do not contain multiplicative squares, i.e., sets of the form {a, ar, as, ars}.


Full version:  pdf,    dvi,    ps,    latex    


(Concerned with sequences A003002 A003003 A003004 A003005 A003022 A003142 A156989 A208746 A259026.)


Received October 18 2014; revised versions received June 12 2015; June 30 2015; July 12 2015. Published in Journal of Integer Sequences, July 16 2015.


Return to Journal of Integer Sequences home page