Number of non-intersecting non-homotopic simple closed curve How many simple closed curves can be put on a orientable genus $g$ surface $\Sigma_g$ such that the following are true:


*

*The curves are pairwise non-homotopic

*The curves are pairwise set-theoretically disjoint.


I am trying to give an upper bound on this number.
 A: This is Lemma 3.2 of this paper by Juvan, Malnic and Mohar.  
Theorem. Let $F$ be a family of non-null homotopic closed curves on a surface $\Sigma$ with $b \geq 0$ boundary components which are pairwise non-homotopic and pairwise disjoint.  Then
$|F| \leq \max (1, 3(g_{\Sigma}-1)+2b)$,
where $g_{\Sigma}$ is the genus of $\Sigma$ (orientable or non-orientable).
A: I am tacitly assuming that your curves should also be non-contractible, otherwise you have to add 1 to the answer. Let's suppose for simplicity that the surface is closed (but the argument adapts in a straightforward way to the non-closed situation) and has negative Euler characteristic.
Clearly, the best solution is to have your curves decompose the surface into pairs-of-pants (discs are forbidden; annuli are forbidden; and any other surface can be further decomposed).
Each pair-of-pants has Euler characteristic -1, and contributes 3/2 boundary curves (dividing by 2 to account for the fact that each boundary curve appears twice). The answer is then
-3/2 the Euler characteristic, as described in the first answer above.
