6'2", 240 and I use 23 cm michelin pro race at 120 psi... (at least that's what the meter reads when I disconnect the hose, it probably drops a bit). No flats in over a year until two weeks ago when I landed in a rut trying to avoid my daughter.
As a bunch of other people said (I"ll repeat because it's important) look for the cause of the flat! YOu can pop your tires from underinflating too.
I think most people will line up their tire label by the valve stem to assist in looking for debris inside the tire. Run your fingers along the inside (careful if you have glass in there).
Also, you may wish to consider using new tubes each time - they aren't that expensive. Forget the cheap glueless patches except as an emergency measure. I carry spare new tubes in my seat pack and I'll patch them at home and reuse them, but I never feel like doing that while I'm riding.
Good luck with your tri