I would suggest a dwarf tall fescue and blue grass mix. Check out
seedland.com, they have a lot of good info. If you have at least 6" of
good soil, it's not compacted, etc, then I'd rent a slice seeder and
use that. Of course, the best time to do this is in the fall, not now.
You're up against a lot of competition from weeds, needing to supply a
lot of water to keep it wet during germination, etc. And it gets worse
in the coming weeks, not better. In late summer/early fall the
conditons are favorable and get better, not worse.

I'd also get the soil tested to make sure the PH is right. In new
construction its not unusual to see it be way off.