Let m \geq 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 a b=c.
Note: a partition of S is a pair of sets A, B such that A \cap B=\emptyset, A \cup B=S.