This is a bit of a soft question, but I am interested in a list of classes of structures (in the sense of model theory) which are "surprisingly" first-order axiomatizable classes. Meaning, the class of structures is defined in such a way that it is not at all obvious that it is in fact first-order axiomatizable. So, for example, the class of fields with exactly 3 elements is axiomatizable, but not surprisingly so.
Examples of classes of structures which are “surprisingly” axiomatizable.
big-listmodel-theorysoft-question
Best Answer
The class of graphs that are 2-colorable.
A graph is 2-colorable if there is an equivalence relation on the set of verteces that has 2cclasses and is such that no adiacent vertexes are in in the same classs.
The description above is second-order, but it is equivalent to the class of graphs without odd cycles, hence first-order axiomatizable.