Improved order-two pumping-length bound

Determine whether the pumping length required for order-two pumping in general finite-memory automata can be reduced substantially below the current Ramsey-type bound.

Background

For an r-register finite-memory automaton with n states, the paper proves that order-two permutations suffice when the pumping length is at most 3(r²!)n. This bound is obtained using a multicolor Ramsey-theoretic argument.

The conclusion identifies reducing this bound substantially as an unresolved question for general finite-memory automata. The question concerns the quantitative dependence of the pumping length on the number of registers and states, not whether order-two pumping is possible at all.