On Kierstead’s conjecture

Abstract

We settle the long-standing Kierstead conjecture in the negative. We do this by constructing a computable linear order with no rational subintervals, where every block has order type finite or ζ \zeta , and where every computable copy has a strongly nontrivial Π 1 0 \Pi ^0_1 automorphism. We also construct a strongly η \eta -like linear order where every block has size at most 4 4 with no rational subinterval such that every Δ 2 0 \Delta ^0_2 isomorphic computable copy has a nontrivial Π 1 0 \Pi ^0_1 automorphism.