论文标题
一些多变量的RADO数字
Some multivariable Rado numbers
论文作者
论文摘要
方程的RADO数是与方程相关的Ramsey理论数量。令$ \ mathcal {e} $为线性方程。用$ \ operatorName {r} _r（\ Mathcal {e}）$最小整数（如果存在），以使任何$ r $ -Coloring $ [1，\ operatotorname {r} {r} _r（r} _r（\ nathcal {e}）$ nure nure nure nure nurs nure nure nurs of monochromantic solocation $ \ by $ {在本文中，我们给出了$ \ sum_ {i = 1}^{m-2} x_i+kx_+kx_ {m-1} = \ ell x_ {m x_ {m} $的$ \ sum_ {i = 1}^{m-2}^{m-2}^{m-2}^{m-2} = \ ell x_ {m} $的上限和下限。此外，我们在$ \ ell = m = 4 $和$ m = 5，\ ell = k+i \（1 \ leq i \ leq 5）$的情况下得出了一些结果。作为概括，线性方程$ \ Mathcal {e} _1，\ Mathcal {e} _2，...，\ Mathcal {E} _r $定义为最小的Integer，如果它存在于任何$ -COLORORON，\ emph {$ r $ -color rado numbers} $ \ mathcal {e} _1，\ Mathcal {e} _2，...，\ Mathcal {e} _r $定义为最小的Integer，如果它存在，则是任何$ r $ r $ -COLORON $ [1，\ operatorname {r} _r（\ Mathcal {e} _1，\ Mathcal {e} _2，...，\ Mathcal {e} _r）] $必须承认单色的解决方案，以对某些$ \ nathcal {e} _i $，$ 1 \ peq c r q r q i q r q。 $ \ operatorName {r} _r（\ Mathcal {e} _1，\ Mathcal {e} _2，...，\ Mathcal {e} _r）$的确切值以及$ \ opperatorname {r} _2 _2 _2（x+y = z，x y = z，x y y x = y 5k） $ \ operatatorName {r} _2（x+y = z，x+a = y）$由lovászlocal lemma给出。
The Rado number of an equation is a Ramsey-theoretic quantity associated to the equation. Let $\mathcal{E}$ be a linear equation. Denote by $\operatorname{R}_r(\mathcal{E})$ the minimal integer, if it exists, such that any $r$-coloring of $[1,\operatorname{R}_r(\mathcal{E})]$ must admit a monochromatic solution to $\mathcal{E}$. In this paper, we give upper and lower bounds for the Rado number of $\sum_{i=1}^{m-2}x_i+kx_{m-1}=\ell x_{m}$, and some exact values are also given. Furthermore, we derive some results for the cases that $\ell=m=4$ and $m=5, \ell=k+i \ (1\leq i\leq 5)$. As a generalization, the \emph{$r$-color Rado numbers} for linear equations $\mathcal{E}_1,\mathcal{E}_2,...,\mathcal{E}_r$ is defined as the minimal integer, if it exists, such that any $r$-coloring of $[1,\operatorname{R}_r(\mathcal{E}_1,\mathcal{E}_2,...,\mathcal{E}_r)]$ must admit a monochromatic solution to some $\mathcal{E}_i$, where $1\leq i\leq r$. A lower bound for $\operatorname{R}_r(\mathcal{E}_1,\mathcal{E}_2,...,\mathcal{E}_r)$ and the exact values of $\operatorname{R}_2(x+y=z,\ell x=y)=5k$ and $\operatorname{R}_2(x+y=z, x+a=y)$ was given by Lovász Local Lemma.
