  This note contains some material promised in our earlier papers on submodel
preservation and the guarded fragment, along with some information on the
current status of the problems mentioned in these papers. Section 1 contains an
early example of failure of Los-Tarski for finite-variable fragments with
binary relations from a 1992 manuscript, with no substantial change of content.
For further background to this example, Section 2 surveys a number of results
concerning the submodel preservation property for various fragments of
first-order logic.
