Let $S\subseteq \{0,\ldots,n\}^d$ be a set of $d$-dimensional vectors of with bounded, natural, coordinates.


We are given that
$$v'+v\_1+\ldots+v\_t=u'+u\_1+\ldots+u\_s$$
where $v\_1,\ldots,v\_t,u\_1,\ldots,u\_s,v',u'\in S$ (and the vectors are not necessarily distinct).


That is, two sets of vectors whose sums are equal.



> 
> I want to prove that, if $t$ and $s$ are large enough, then there
> exist subsets $I\subseteq \{1,\ldots,t\}$ and $J\subseteq \{1,\ldots
> > s\}$ such that $$\sum\_{i\in I}v\_i=\sum\_{j\in J}u\_j$$
> 
> 
> 


Note that, without the assumption of the equal sum above, there may not be such sets (e.g., if all the $v\_i$ are $(1,0)$, and all the $u\_i$ are $(0,1)$). Also, for small $s,t$ there may not be such sets.


**Some informal thoughts:**
My intuition is that for large enough $s,t$, we can force a lot of repetitions of vectors within the sets, and then we can ``tailor'' equal sums. This is somewhat akin to a vector version of Erdős-Ginzburg-Ziv (or the Van Emde Boas - Kruyswijk variation, which looks at vectors), but instead of looking at the finite abelian group, I have the sum above to bound the behaviour.


Also, I don't really care about tight bounds for $s,t$. They can be as large as needed (e.g., exponential, or even double exponential in $|S|,n$ is fine).



Equal subset-sums of bounded vectors
As I understand, your $n$ and $d$ are fixed, and you want to prove the existence of corresponding non-empty subsets $I$, $J$ provided that both $t$, $s$ are large enough (greater than some constant $C(n,d)$).


This is true.


We find a positive integer $M$ satisfying the following condition: whenever $U,V$ are finite subsets of $\{0,\ldots,n\}^d$ such that their conic hulls (convex cones generated by $U,V$ respectively) have a common non-zero point, we get $\sum\_{i=1}^{M\_1} v\_i=\sum\_{i=1}^{M\_2} u\_i$ for certain $M\_1,M\_2\in \{1,\ldots,M\}$ and $v\_i\in V$, $u\_i\in U$ (not necessarily distinct).


Now assume that $t,s$ are large enough and define $$V=\{v\_i\,\text{which appear at least}\, M\,\text{times between}\,v\_1,\ldots,v\_t \},\\U=\{u\_i\,\text{which appear at least}\, M\,\text{times between}\,u\_1,\ldots,u\_s \}.$$
If the conic hulls of $V$, $U$ have non-empty intersection, we are done. Otherwise by Hahn--Banach the sets $U$, $V$ are separated by a linear functional $f:\mathbb{R}^d\to \mathbb{R}$: $f(u)>0>f(v)$ for all non-zero $u\in U$, $v\in V$. This $f$ depends on $U$, $V$, fix it for each pair $U,V$. If both $U$, $V$ contain a zero vector, again we are done. Otherwise applying $f$ to both parts of the equality $v'+v\_1+\ldots+v\_t=u'+u\_1+\ldots+u\_s$ we see that LHS is much less than RHS if $t$, $s$ are large enough (because the number of $i$ for which $u\_i\notin U$ or $v\_i\notin V$ is bounded, and for other indices either $f(u\_i)$ or $f(v\_i)$ is bounded away from 0.)


