Let $n\in \mathbb{N}$, $R$ be a binary relation on $[n]$, and$C_1(i,j),\ldots,C_n(i,j) \in \mathbb{Z}$, for $i,j \in [n]$. We define theexponential system of equations $\mathcal{E}(R,(C_k(i,j)_{i,j,k})$ to be thesystem \[ X_i^{Y_1^{C_1(i,j)} \cdots Y_n^{C_n(i,j)} } = X_j , \text{ for }(i,j) \in R ,\] in variables $X_1,\ldots,X_n,Y_1,\ldots,Y_n$. The aim of thispaper is to classify precisely which of these systems admit a monochromaticsolution ($X_i,Y_i \not=1)$ in an arbitrary finite colouring of the naturalnumbers. This result could be viewed as an analogue of Rado's theorem forexponential patterns.
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机译：设$ \ n \ mathbb {N} $中的$ R $为$ [n] $上的二元关系，以及$ ma_1，\ ldots，C_n（i，j）\ in \ mathbb {Z } $，表示[i] j中的$ i，j。我们将方程式$ \ mathcal {E}（R，（C_k（i，j）_ {i，j，k}）$的指数系统定义为系统\ [X_i ^ {Y_1 ^ {C_1（i，j）} \ cdots Y_n ^ {C_n（i，j）}} = X_j，\ text {for}（i，j）\ in R，\]在变量$ X_1，\ ldots，X_n，Y_1，\ ldots，Y_n $中。本文的目的是精确分类这些系统中的哪一个在自然数的任意有限着色下允许单色解（$ X_i，Y_i \ not = 1）$，这一结果可以看做是Rado定理的指数模式的类似物。
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