  We study the satisfiability problem for the fluted fragment extended with
transitive relations. The logic enjoys the finite model property when only one
transitive relation is available and the finite model property is lost when
additionally either equality or a second transitive relation is allowed. We
show that the satisfiability problem for the fluted fragment with one
transitive relation and equality remains decidable. On the other hand we show
that the satisfiability problem is undecidable already for the two-variable
fragment of the logic in the presence of three transitive relations (or two
transitive relations and equality).
