R.E. Allen, Plato's Parmenides: Translation and Analysis (Minnesota 1983).
The classic treatment of this argument is Gregory Vlastos' The "Third Man Argument in the Parmenides," in R.E. Allen, ed., Studies in Plato's Metaphysics, 231-63; originally published in Philosophical Review 63 (1954), 319-49.
A more recent discussion of both regress arguments, with extensive notes and bibliography, may be found in Gail Fine, On Ideas: Aristotle's Criticisms of Plato's Theory of Forms (Oxford, 1993), pp. 204-224.
Uniqueness: (U)
There is exactly one form corresponding to every predicate that has a form.
Self-Predication (SP)
Any form can be predicated of itself. (Largeness is large).
Non-Identity (NI)
If a thing has a certain character, it cannot be identical with the form in virtue of which we apprehend that character.
One-Over-Many (OM)
If there is a set of things with one predicate, then there is one form for that predicate, and the members of the set have that predicate in virtue of that form.
1. Let there be a Set #1 of several A's. (There are several large things.)
2. (From OM). The A's participate in one and the same form, F(A). Call it Form1(A).
3. (From U) There is one and only one F(A). Form1(A) is the only F(A).
4. (From SP) Form1(A) is itself A. (Largeness #1 is itself large.)
5. (From NI) Form1(A) is not itself a member of Set #1.
6. Since Form1(A) is large, it and set #1 together form a new Set #2 of A's (There is a new set of large things including the Form of Large).
7. (From (OM) The new Set #2 of A things all participate in one and the same form . Call it Form2(A). (A "new" form of Largness.)
8. (From SP) Form2(A) is itself A. (Largeness #2 is itself large.)
9. (From NI) (repeating step 5) Form2 (A) is not itself a member of Set #2.
10. (Repeating step 6.) There is a new set #3 of A's.
11. (Repeating step 7.) The new set #3 of As participates in a new Form3 (A).
12. There is an infinite regress of forms of (A). (or of largeness).
13. Therefore, it is not the case that U (forms are not unique).
Comment: The argument is a reductio leading to a contradiction.