Geodesic
Sets of natural numbers with proscribed subsets
Journal of integer sequences, Tome 18 (2015) no. 7.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let ? be a set of subsets of the natural numbers, and let $G_{\?}(n)$ be the maximum cardinality of a subset of ${1, 2, . . . , n}$ that does not have any subsets that are in ?. We consider the general problem of giving upper bounds on $G_{\?}(n)$, and give new results for some ? 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 ${a, ar, as, ars}$.
Classification : 11B05, 11B25, 11B75, 11B83, 05D10
Keywords: geometric progression-free sequence, Ramsey theory 
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     author = {O'Bryant, Kevin},
     title = {Sets of natural numbers with proscribed subsets},
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     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_7_a1/}
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O'Bryant, Kevin. Sets of natural numbers with proscribed subsets. Journal of integer sequences, Tome 18 (2015) no. 7. http://geodesic.mathdoc.fr/item/JIS_2015__18_7_a1/