  Working with uncountable structures of fixed cardinality, we investigate the
complexity of certain equivalence relations and show that if V = L, then many
of them are \Sigma^1_1-complete, in particular the isomorphism relation of
dense linear orders.
  Then we show that it is undecidable in ZFC whether or not the isomorphism
relation of a certain well behaved theory (stable, NDOP, NOTOP) is
\Sigma^1_1-complete (it is, if V = L, but can be forced not to be).
