On the degree of regularity of generalized van der Waerden triples

Nikos Frantzikinakis, Bruce Landman, Aaron Robertson

Research output: Contribution to journalArticle


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
Issue number1
Publication statusPublished - Jul 1 2006
Externally publishedYes


ASJC Scopus subject areas

  • Applied Mathematics

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