If you count bordreaux, maroon, etc. as shades of red, I think I can do it in six. Gold/yellow, white, blue, red, green and brown. (Gamma Phi screws the whole thing up.
) Of course, if you want to get technical, pink is really just a member of the red family, and in that case we can do away with my "green" category, since I think the only sorority that doesn't have gold/yellow, white, blue, brown or some shade of red is DZ.