KöMaL problem A, 2022

For every $i \in \mathbb{N}$ let $A_i$, $B_i$ and $C_i$ be three finite and pairwise disjoint subsets of $\mathbb{N}$. Suppose that for every partition of $\mathbb{N}$ consisting of sets $A$, $B$ and $C$ there exists $i\in \mathbb{N}$ such that $A_i \subset A$, $B_i \subset B$ and $C_i \subset C$. Prove that there also exists a finite $S\subset \mathbb{N}$ such that for every partition of $\mathbb{N}$ consisting of sets $A$, $B$ and $C$ there exists $i\in S$ such that $A_i \subset A$, $B_i \subset B$ and $C_i \subset C$.