jmc

We claim that $243$ is the minimal value of $m$. Let the two partitioned sets be $A$ and $B$; we will try to partition $3,9,27,81,$ and $243$ such that the $ab=c$ condition is not satisfied. Without loss of generality, we place $3$ in $A$. Then $9$ must be placed in $B$, so $81$ must be placed in $A$, and $27$ must be placed in $B$. Then $243$ cannot be placed in any set, so we know $m$ is less than or equal to $243$. For $m\le 242$, we can partition $S$ into $S\cap \{3,4,5,6,7,8,81,82,83,\mathrm{84...242}\}$ and $S\cap \{9,10,\mathrm{11...80}\}$, and in neither set are there values where $ab=c$ (since $8<(3\text{ to }8{)}^2<81$ and $(9\text{ to }80{)}^2>80$). Thus $m=\boxed{243}$.