Geodesic
Sets of natural numbers with proscribed subsets
Journal of integer sequences, Tome 18 (2015) no. 7
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 
@article{JIS_2015__18_7_a1,
     author = {O'Bryant,  Kevin},
     title = {Sets of natural numbers with proscribed subsets},
     journal = {Journal of integer sequences},
     year = {2015},
     volume = {18},
     number = {7},
     zbl = {1393.11005},
     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/