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}.
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(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.
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