Two structures with identical one-variable definable sets can differ in pseudo-o-minimality, refuting an axiomatization by one-variable conditions and a pigeonhole principle conjecture.
Locally o-minimal structures and structures with locally o-minimal open core
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Pseudo-finite sets, pseudo-o-minimality
Two structures with identical one-variable definable sets can differ in pseudo-o-minimality, refuting an axiomatization by one-variable conditions and a pigeonhole principle conjecture.