OIM 2021 Problem 1

Let $P = \{p_1,p_2,\ldots, p_{10}\}$ be a set of $10$ different prime numbers and let $A$ be the set of all the integers greater than $1$ so that their prime decomposition only contains primes of $P$ . The elements of $A$ are colored in such a way that:

• each element of $P$ has a different color,
• if $m,n \in A$ , then $mn$ is the same color of $m$ or $n$ ,
• for any pair of different colors $\mathcal{R}$ and $\mathcal{S}$ , there are no $j,k,m,n\in A$ (not necessarily distinct from one another), with $j,k$ colored $\mathcal{R}$ and $m,n$ colored $\mathcal{S}$ , so that $j$ is a divisor of $m$ and $n$ is a divisor of $k$ , simultaneously.

Prove that there exists a prime of $P$ so that all its multiples in $A$ are the same color.