TY - JOUR
T1 - On the degree of regularity of generalized van der Waerden triples
AU - Frantzikinakis, Nikos
AU - Landman, Bruce
AU - Robertson, Aaron
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
SN - 0196-8858
VL - 37
SP - 124
EP - 128
JO - Advances in Applied Mathematics
JF - Advances in Applied Mathematics
IS - 1
ER -