Polynomial orbit growth and monadic NIP

Establish whether, for countable homogeneous omega-categorical relational structures, subset-orbit growth bounded by 2^{O(n)} is equivalent to monadic NIP, meaning that NIP persists under arbitrary expansions by unary predicates.

Background

The paper places finite-on-top-of-temporal structures within a broader hierarchy based on the growth rate of automorphism-group orbits on finite subsets. It reports a conjectured equivalence for countable homogeneous omega-categorical relational structures: an exponential upper bound on subset-orbit growth should characterize monadic NIP. Only one implication is known according to the paper, so the converse remains unresolved.

References

For countable homogeneous \omega-categorical relational structures, growth bounded by $2{O(n)}$ is conjectured to be equivalent to monadic NIP, meaning that NIP persists under arbitrary expansions by unary predicatesConjecture~1.

— Finite-on-Top-of-Temporal CSPs in Fixed-Point Logic  (2610.18287 - Bodirsky et al., 13 Oct 2026) in Section 1, Introduction, subsection "Restricted Orbit Growth"