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Dec 7, 2020 at 0:49 comment added Zsbán Ambrus @j.c. link to that Erich Friedman survey article is now erich-friedman.github.io/papers/squares/squares.html
Mar 27, 2010 at 21:18 vote accept Anton Geraschenko
Mar 27, 2010 at 21:17 comment added Anton Geraschenko All these references are excellent! In particular, Nagamochi's Packing Unit Squares in a Rectangle (available at combinatorics.org/Volume_12/PDF/v12i1r37.pdf) proves that an $(a_1-\varepsilon)\times(a_2-\varepsilon)$ rectangle can hold at most $a_1a_2-2$ unit squares. The results in the references suggest that this is as good an answer as I'm likely to get, so I'm going to accept it.
Mar 27, 2010 at 10:36 comment added j.c. One source with plenty of pictures is Erich Friedman's website, which includes the following survey: www2.stetson.edu/~efriedma/papers/squares/squares.html
Mar 27, 2010 at 6:54 history answered Gerry Myerson CC BY-SA 2.5