On the degree of regularity of generalized van der Waerden triples

Nikos Frantzikinakis, Bruce M. Landman, Aaron Robertson

Research output: Contribution to journalArticle

Abstract

Let 1 {less-than or slanted equal to} a {less-than or slanted equal to} b be integers. A triple of the form ( x, a x + d, b x + 2 d ), where x, d are positive integers is called an ( a, b )-triple. The degree of regularity of the family of all ( a, b )-triples, denoted dor ( a, b ), is the maximum integer r such that every r-coloring of N admits a monochromatic ( a, b )-triple. We settle, in the affirmative, the conjecture that dor ( a, b ) < ∞ for all ( a, b ) ≠ ( 1, 1 ). We also disprove the conjecture that dor ( a, b ) ∈ { 1, 2, ∞ } for all ( a, b ).

Original languageEnglish (US)
Pages (from-to)124-128
Number of pages5
JournalAdvances in Applied Mathematics
Volume37
Issue number1
DOIs
StatePublished - Jul 1 2006

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On the degree of regularity of generalized van der Waerden triples. / Frantzikinakis, Nikos; Landman, Bruce M.; Robertson, Aaron.

In: Advances in Applied Mathematics, Vol. 37, No. 1, 01.07.2006, p. 124-128.

Research output: Contribution to journalArticle

Frantzikinakis, Nikos ; Landman, Bruce M. ; Robertson, Aaron. / On the degree of regularity of generalized van der Waerden triples. In: Advances in Applied Mathematics. 2006 ; Vol. 37, No. 1. pp. 124-128.
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