Suppose we are given a positive integer $k$. Let $K_k$ denote the complete (undirected and simple) graph with vertices $1, 2, \dots, k$. The set of edges of $K_k$ is the set $E_k = \{ \{ x,y \} \mid \ 1 \leq x < y \leq k\}$.

A valuation of $K_k$ is a function $\omega: E_k \rightarrow \mathbb Z$. A splitting of $K_k$ is a partition $\{ 1, 2, \dots, k \} = A \cup B$ of the vertices of $K_k$ into two nonempty sets $A$ and $B$.

Given a valuation $\omega$ of $K_k$ and a positive integer $n$, we call a splitting $A \cup B$ of $K_k$ $n$-valid, if the number $$\sum_{(x,y) \in A \times B} \omega(\{ x,y \})$$ is divisible by $n$.

Using Ramsey's theorem, one can prove that for every positive integer $n$, there exists a positive integer $k$ such that the following condition holds:

For any valuation $\omega$ of $K_k$, there is a splitting of $K_k$ that is $n$-valid.

If there exists at least one, there has to exist a smallest $k$ satisfying the above condition which we denote by $\eta(n)$.

Using the Combinatorial Nullstellensatz from N. Alon, one can show that $\eta(p) = 2p$ for odd primes $p$.

I now want to know if $\eta(n) = 2n$ is true for every integer $n \geq 3$.