TITLE: forcing square with small conditions
QUESTION [5 upvotes]: In the paper, Large cardinals and definable counterexamples to the continuum hypothesis, Foreman and Magidor mention a way to force $\square_{\omega_1}$ with countable conditions.  (This is used in Theorem 2.17.)  It not hard to guess what this is and prove it is countably closed and $\omega_2$-c.c. under CH.
Questions:
(1) Are a definition of this forcing and proofs of its basic properties explicitly written up anywhere?
(2) Does this generalize to higher cardinals, for example forcing $\square_{\omega_2}$ with $\omega_1$-sized conditions?  It seems complications could arise.

REPLY [5 votes]: The appropriate reference for a forcing for adding a square sequence with countable conditions is: "S-forcing I. A "black-box" theorem for morasses, with applications to super-Souslin trees", Shelah and Stanley, Israel J. Math, vol. 43, no. 3, 1982. See section 4 example 1 pages 203-204.