About a year and a half ago, Schroeppel and Hickerson found a bunch of patterns with very small predecessor counts. These are used as modules in existence proofs for GOE's within various rectangles, some quite cozy. 1991 March 28, Bill Gosper, “on Life (and Death)”, in comp.theory.cell-automata (Usenet):
This describes briefly the discovery of the first GOL GOE by Roger Banks, saying: "[he] used sophisticated mathematical techniques to prove that a certain 9-by-33 rectangular pattern is itself a Garden-of-Eden pattern." 1999 April 1, Tim Tyler, “proof for garden of eden in GOL ?”, in comp.theory.cell-automata (Usenet):
As far as I know, even a definite result for the 6x6 case would be progress. The most recent investigation of GoEs that I know about was some interesting work that Donald Knuth recently did in passing for his upcoming Volume 4 of _The Art of Computer Programming_, using binary decision diagrams. 2008 October 22, Dave Greene, “State wrt "Garden of Eden"in Conway's "Game of Life"”, in comp.theory.cell-automata (Usenet):
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(abbreviation, alt-of, initialism, qualifier:earth science) Initialism of Great Oxygenation Event.