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Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 220)

Abstract

Two well-studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression or in geometric progression. We study generalizations of this problem by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence \(x, f_n(x), f_n(f_n(x))\), where \(f_n(x) = x + n\), n a nonzero integer. Thus, avoiding 3-term arithmetic progressions are equivalent to containing no three elements of the form \(x, f_n(x), f_n(f_n(x))\) with \(f_n \in \mathscr {F}_\mathrm{t}\), the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions (\(f_n(x) = nx\) with \(n > 1\) a natural number) and exponential progressions (\(f_n(x) = x^n\)). Progression-free sets are often constructed “greedily,” including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1 and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density; however, we consider several different notions of density, including the uniform and exponential densities.

Keywords

  • Ramsey theory
  • Progressions

This work supported in part by NSF Grants DMS1265673, DMS1561945, DMS1347804 and Williams College.

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Acknowledgements

This research was conducted as part of the 2014 SMALL REU program at Williams College and was supported by NSF grants DMS1265673, DMS1561945, DMS1347804, Williams College, and the Clare Boothe Luce Program of the Henry Luce Foundation. It is a pleasure to thank them for their support.

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Correspondence to Steven J. Miller .

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Best, A. et al. (2017). Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_4

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