  We study the satisfiability problem for the two-variable first-order logic
over structures with one transitive relation. % We show that the problem is
decidable in 2-NExpTime for the fragment consisting of formulas where
existential quantifiers are guarded by transitive atoms. As this fragment
enjoys neither the finite model nor the tree model property, to show
decidability we introduce novel model construction technique based on the
infinite Ramsey theorem.
  We also point out why the technique is not sufficient to obtain decidability
for the full two-variable logic with one transitive relation, hence contrary to
our previous claim, [FO$^2$ with one transitive relation is decidable, STACS
2013: 317-328], the status of the latter problem remains open.
