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Generalization of Dickson's Lemma

Given $\{v^i\}_{i \in \mathbb{N}} \subseteq \mathbb{N}^n$, and $\cup_{k=1, \ldots, m} C_j = \mathbb{N}^n$ for some $m$, where each $C_k$ is a cone generated by rational vectors. My question is: does ...