Abstract
Let m, a, b be positive integers, with \(\gcd (a,b)=1\). The disjunctive Rado number for the pair of equations \(y-x=ma\), \(y-x=mb\), is the least positive integer \(R={\mathscr {R}}_d(ma,mb)\), if it exists, such that every 2-coloring \(\chi\) of the integers in \(\{1,\ldots ,R\}\) admits a solution to at least one of \(\chi (x)=\chi (x+ma)\), \(\chi (x)=\chi (x+mb)\). We show that \({\mathscr {R}}_d(ma,mb)\) exists if and only if ab is even, and that it equals \(m(a+b-1)+1\) in this case. We also show that there are exactly \(2^m\) valid 2-colorings of \([1,m(a+b-1)]\) for the equations \(y-x=ma\) and \(y-x=mb\), and use this to obtain another proof of the formula for \({\mathscr {R}}_d(ma,mb)\).
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18 July 2022
The original version is updated due to minor errors and spacing issues.
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Dileep, A., Moondra, J. & Tripathi, A. New Proofs for the Disjunctive Rado Number of the Equations \(x_1-x_2=a\) and \(x_1-x_2=b\). Graphs and Combinatorics 38, 38 (2022). https://doi.org/10.1007/s00373-021-02400-y
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DOI: https://doi.org/10.1007/s00373-021-02400-y