TY - JOUR

T1 - On the degree of regularity of generalized van der Waerden triples

AU - Frantzikinakis, Nikos

AU - Landman, Bruce

AU - Robertson, Aaron

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2006/7

Y1 - 2006/7

N2 - 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 ).

AB - 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 ).

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U2 - 10.1016/j.aam.2005.08.003

DO - 10.1016/j.aam.2005.08.003

M3 - Article

AN - SCOPUS:33646511994

VL - 37

SP - 124

EP - 128

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

IS - 1

ER -