There is an ubiquitous pattern of questions concerning assumedly any kind of mathematical object or structure: groups, graphs, numbers, categories, and so on. It goes like this (informally):

Can a class of objects or structures of a given kind

Xthat is characterized by some "external condition"Ybe defined by a conditionZin their respective "internal" language, and if so: how?

Well-known examples ("external condition" **=** "internal condition"):

groups $G$ isomorphic to a subgroup of the symmetric group on $G$

**=**all groups (*Cayley's theorem*)graphs embeddable in the plane

**=**graphs not containing a subgraph that is a subdivision of $K_5$ or $K_{3,3}$ (*Kuratowski's theorem*)numbers

*n*of trees on*k*labeled vertices**=**numbers*n = k*^{k-2}for some*k*> 1 (*Cayley's theorem on trees*)numbers

*n*with only one group of order*n***=**numbers*n = p*for some_{1}· p_{2}· ... · p_{k}*k*> 0, where the*p*are distinct primes and no_{i}*p*is divisible by any_{j}-1*p*(_{i}*cyclic numbers*, see Sloane's A003277)

Further examples from MO:

Can we recognize when a category is equivalent to the category of models of a first order theory?

Can you determine whether a graph is the 1-skeleton of a polytope?

Question #1:What's the proper way to characterize this pattern of questions? What's the common context / rationale?

Question #2:How is the introductory question to be posed properly?