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Forced differences between terms of subsequences of integer sequences


Authors: Michael Gilpin and Robert Shelton
Journal: Proc. Amer. Math. Soc. 88 (1983), 569-578
MSC: Primary 10L10
DOI: https://doi.org/10.1090/S0002-9939-1983-0702277-1
MathSciNet review: 702277
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Abstract: Let $ {a_1},{a_2}, \ldots $ be a sequence of integers and let $ D = \left\{ {{d_1}, \ldots ,{d_k}} \right\}$ be a fixed finite set of integers. For each positive integer $ n$ we investigate the problem of choosing maximal subsequences $ {a_{{i_1}}}, \ldots ,{a_{{i_t}}}$ from $ {a_1}, \ldots ,{a_n}$ such that $ \left\vert {{a_{{i_\alpha }}} - {a_{{i_\beta }}}} \right\vert \notin D$ for $ \alpha \ne \beta $. An asymptotic form for $ t$, the maximum length of such subsequences, is derived in the special case $ {a_i} = i$.


References [Enhancements On Off] (What's this?)

  • [1] Michael Gilpin and Robert Shelton, Problem 1152, Math. Mag. 55 (1982), 237.
  • [2] Chung Laung Liu, Elements of discrete mathematics, McGraw-Hill Book Co., New York-Auckland-Bogotá, 1977. McGraw-Hill Computer Science Series. MR 0520385
  • [3] -, Topics in combinatorial mathematics, Math. Assoc. Amer., Washington, D.C., 1972.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0702277-1
Keywords: Subsequences, blocks, pigeonhole principle
Article copyright: © Copyright 1983 American Mathematical Society

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