How do I know if we have mold?

My husband and I live in an apartment building and our upstairs neighbors' bathroom has overflowed several times and each time, due to cracks in their linoleum, water has seeped through their floor and through our bedroom ceiling. The landlord says that we don't have mold but there are dark water stains all over our ceiling by our light fixture where the water leaked through. I'm afraid of mold because they haven't looked inside the ceiling at all, they just said to let it dry out and call it a day. How do we know if there is mold in the ceiling and how do we get rid of it? The landlord's maintenance guy said we could paint the ceiling and that would kill any mold, but I don't believe him. Anybody have any insight into this?