Question 12
Let $m \ge 3$ be an integer and let $S = \{3, 4, 5, \ldots, m\}.$ Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a,$ $b,$ and $c$ (not necessarily distinct) such that $ab = c.$
Note: a partition of $S$ is a pair of sets $A,$ $B$ such that $A \cap B = \emptyset$ and $A \cup B = S.$
Note: a partition of $S$ is a pair of sets $A,$ $B$ such that $A \cap B = \emptyset$ and $A \cup B = S.$