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Disjunctive Rado numbers for \(x_1+x_2+c=x_3\). (English) Zbl 1121.05120

Summary: Given two equations \(E_1\) and \(E_2\), the disjunctive Rado number for \(E_1\) and \(E_2\) is the least integer \(n\), provided that it exists, such that for every coloring of the set \(\{1,2,\dots,n\}\) with two colors there exists a monochromatic solution to either \(E_1\) or \(E_2\). If no such integer \(n\) exists, then the disjunctive Rado number for \(E_1\) and \(E_2\) is infinite. Let \(R(c,k)\) represent the disjunctive Rado number for the equations \(x_1+x_2+c=x_3\) and \(x_1+x_2+k=x_3\). In this paper the values of \(R(c,k)\) are found for all natural numbers \(c\) and \(k\) where \(c\leq k\). It is shown that \[ R(c,k)= \begin{cases} 4c+5 &\text{if}\quad c\leq k\leq c+1\\ 3c+4 &\text{if}\quad c+2\leq k \leq 3c+2\\ k+2 &\text{if}\quad 3c+3\leq k\leq 4c+2\\ 4c+5 &\text{if}\quad 4c+3\leq k.\end{cases} \]

MSC:

05D10 Ramsey theory
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