ruletheworld,
The majors won't ever be handing out greencards. They each have about 13,000 resumes on file, and its extremely competitive to get an interview. The number of pilots being hired without a 4 year degree is extremely low. Ive heard of one in the past couple of years.
I know of some without a degree who have gone to Atlas, but in order to get on at United, Delta, American, Southwest, or Fedex it would be foolhardy not to get one.
FOs are also only making up about 10% of new hire classes and usually have other things on their resume to overcome the lack of PIC time. (union work, volunteering, extensive networking, etc.)
Not trying to be discouraging, just presenting the reality. Regionals are a different story however.