Does anyone know any reference or proof for the following problem?



> 
> Let $m$ and $n$ be positive integers, $m,n \geq 2$. Each positive integer is coloured in one of $m$ different colours. Is it possible to find $n$ positive integers $x\_1, x\_2, ..., x\_n$ such that all positive numbers of the form $$\sum\_{i=1}^n \epsilon\_ix\_i,$$ where $\epsilon\_i \in \{0,1\},$ have the same colour, when:
> 
> 
> a) $n \leq m$?  
> 
> b) $n>m$?
> 
> 
> 


Any help would be appreciated.



Colouring Positive Integers
Yes. This is a weak form of Hindman's theorem. See, for example, <https://en.wikipedia.org/wiki/IP_set> .


