正则非齐次方程的Rado数
Rado Numbers of Regular Nonhomogeneous Equations
摘要：
我们考虑正则方程$\mathcal{E}(b)$的Rado数形式为\[ c_1x_1+c_2x_2+\dots+ c_{k-1}x_{k-1} = x_k + b, \]其中$b \in \mathbb{Z}$和$c_i \in \mathbb{Z}^{+}$对所有$i$ .给出了$b>0$和$b<0$的$t$ -色Rado数$r(\mathcal{E}(b);t)$用$r(\mathcal{E}(0);t)$表示的上界和下界的充分条件.我们还给出了例子，从这些结果中得到的Rado数的精确值。
英文摘要：
We consider Rado numbers of the regular equations $\mathcal{E}(b)$ of the form \[ c_1x_1+c_2x_2+\dots+ c_{k-1}x_{k-1} = x_k + b, \] where $b \in \mathbb{Z}$ and $c_i \in \mathbb{Z}^{+}$ for all $i$. We give the upper bounds and the sufficient condition for the lower bounds for $t$-color Rado numbers $r(\mathcal{E}(b);t)$ in terms of $r(\mathcal{E}(0);t)$ for both $b>0$ and $b<0$. We also give examples where the exact values of Rado numbers are obtained from these results.
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