Lucy wants an “Italian haircut”, a shorter style. The relevant portion of the clip begins at about 2:10 and ends at 3:00.
Transcript:
Lucy: You know what I’m going to do? I’m going to get an Italian haircut.
Ricky: Oh no you’re not!
Lucy: Why not?
Ricky: Because I like your hair the way it is.
Lucy: But it would look so good short. Please?
Ricky: Look… for my son’s sake, no.
Lucy: What do you mean for your son’s sake?
Ricky: I’ll ‘splain.
Lucy: Ok. ‘Splain.
Ricky: All people in the world are divided into two groups… men and women.
Lucy: [sarcastically] I know. It’s a wonderful arrangement.
Ricky: Now. Men have short hair, and women have long hair. That’s the difference between them.
Lucy: Oh?
Ricky: Now, I don’t want my son to be confused. He should know whether he should call you mother or father.
Lucy: [walking away] Oh… men! You make me sick.
And later at 13:45 in the video…
Ethel: Well, it’s a terrible thing to say about anyone, but I guess Ricky’s just a man.