Ramsey functions for quasi-progressions

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Abstract

A quasi-progression of diameter n is a finite sequence {x1 , . . . , xk} for which there exists a positive integer L such that L ≤ xi - xi-1 ≤ L + n for i = 2, . . . , k. Let Qn(k) be the least positive integer such that every 2-coloring of {1 , . . . , Qn(k)} will contain a monochromatic k-term quasi-progression of diameter n. We give a lower bound for Qk-i(k) in terms of k and i that holds for all k > i ≥ 1. Upper bounds are obtained for Qn(k) in all cases for which n ≥ [2k/3]. In particular, we show that Q[2k/3](k) ≤ 43/324 k3(1 + o(1)). Exact formulae are found for Qk-1(k) and Qk-2(k). We include a table of computer-generated values of Qn(k), and make several conjectures.

Original languageEnglish (US)
Pages (from-to)131-142
Number of pages12
JournalGraphs and Combinatorics
Volume14
Issue number2
DOIs
StatePublished - Jan 1 1998
Externally publishedYes

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

Cite this

Ramsey functions for quasi-progressions. / Landman, Bruce M.

In: Graphs and Combinatorics, Vol. 14, No. 2, 01.01.1998, p. 131-142.

Research output: Contribution to journalArticle

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