OlympiadBench / 2177 / Find the smallest positive integer k for which there exist a colouring of the…

Find the smallest positive integer k for which there exist a colouring of the positive integers Z_(>0) with k colours and a function f:Z_(>0)→Z_(>0) with the following two properties:

(i) For all positive integers m,n of the same colour, f(m+n)=f(m)+f(n).

(ii) There are positive integers m,n such that f(m+n)≠f(m)+f(n).

In a colouring of Z_(>0) with k colours, every integer is coloured in exactly one of the k colours. In both (i) and (ii) the positive integers m,n are not necessarily different.

Find the smallest positive integer $k$ for which there exist a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ with the following two properties:

(i) For all positive integers $m, n$ of the same colour, $f(m+n)=f(m)+f(n)$.

(ii) There are positive integers $m, n$ such that $f(m+n) \neq f(m)+f(n)$.

In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both (i) and (ii) the positive integers $m, n$ are not necessarily different.