4 problems

The model-theoretic classification conjecture for finitely homogeneous structures
arXiv@arxiv
Classification conjecture. Trace-equivalence classes of finitely homogeneous structures should correspond to natural model-theoretic properties.
Conjectureopenhomogeneous-structures model-theory trace-equivalence
1
The countability conjecture for finitely homogeneous structures
arXiv@arxiv
Countability conjecture. There are $\aleph_0$ finitely homogeneous structures up to trace equivalence.
Conjectureopenhomogeneous-structures model-theory trace-equivalence
0
The strong conjecture on finitely homogeneous structures up to trace equivalence
arXiv@arxiv
Strong conjecture. For every $k$ there are only finitely many finitely homogeneous structures in a $k$ -ary language up to trace equivalence.
Conjectureopenhomogeneous-structures model-theory trace-equivalence
0
The positive-pair conjecture for SL_n-trace equivalence
arXiv@arxiv
Let $F_r$ be a free group, and call two pairs of words $(u,v)$ and $(u',v')$ $\operatorname{SL}_n$ -trace equivalent when the corresponding words have equal traces under every repre…
Conjectureopenfree-groups representation-varieties trace-equivalence
0

The model-theoretic classification conjecture for finitely homogeneous structures