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 there exist $i,j \in \mathbb{N}$ and $k=1, \ldots, m$, such that $v^i, v^j, v^j - v^i \in C\_k?$


For example, when $m=1$, this statement is correct since it reduces to the Dickson's Lemma.



Generalization of Dickson's Lemma
Yes.


The condition that the union of the $C\_k$ is $\mathbb N^n$ is not strictly necessary - all you need is that each $v^i$ is in some $C\_k$.


Indeed, there must be some $k$ such that infinitely many $v^i$ lie in $C\_k$.


By Gordan's lemma, $C\_k$ is generated (as a semigroup) by finitely many vectors in $C\_k$. Call these $w\_1,\dots, w\_r$. Write $v^i= \sum\_{t=1}^r a\_{i,t}w\_t$ for all $i$ such that $v^i \in C\_k$ - we may have multiple choices of $a\_{i,t}$, but just pick one.


Now apply Dickson's lemma to the $a\_{i,t}$, obtaining $i$ and $j$ with $v^i\in C\_k, v^j \in C\_k$, and $a\_{i,t} \leq a\_{j,t}$ for all $t$. It follows that $v^j - v^i = \sum\_{t=1}^r (a\_{j,t} - a\_{i,t}) w-t \in C\_k$, as desired.


