Two Formulas of 2-Color Off-Diagonal Rado Numbers

Let ε 0 , ε 1 be two linear homogenous equations, each with at least three variables and coefficients not all the same sign. Define the 2-color off-diagonal Rado number R 2 ( ε 0 , ε 1 ) to be the smallest integer N such that for any 2-coloring of [1, N ], it must admit a monochromatic solution to ε...

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Bibliographic Details
Published inGraphs and combinatorics Vol. 31; no. 1; pp. 299 - 307
Main Authors Yao, Olivia X. M., Xia, Ernest X. W.
Format Journal Article
LanguageEnglish
Published Tokyo Springer Japan 01.01.2015
Springer Nature B.V
Subjects
Color
Combinatorial analysis
Combinatorics
Engineering Design
Graphs
Integers
Mathematical analysis
Mathematics
Mathematics and Statistics
Original Paper
Texts
Schur number
11B75
Off-diagonal Rado number
05D10
05C55
Rado number
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Summary:Let ε 0 , ε 1 be two linear homogenous equations, each with at least three variables and coefficients not all the same sign. Define the 2-color off-diagonal Rado number R 2 ( ε 0 , ε 1 ) to be the smallest integer N such that for any 2-coloring of [1, N ], it must admit a monochromatic solution to ε 0 of the first color or a monochromatic solution to ε 1 of the second color. In this paper, we establish two exact formulas of R 2 (3 x + 3 y =  z , 3 x + 3 qy =  z ) and R 2 (2 x + 3 y =  z , 2 x + 2 qy =  z ).
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 23
ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-013-1378-9