Labelled growth rates of $\omega$-categorical structures and applications in choiceless set theory

We study the labelled growth rate of an $\omega$-categorical structure $\mathfrak{A}$, i.e., the number of orbits of $Aut(\mathfrak{A})$ on $n$-tuples of distinct elements, and show that the model-theoretic property of monadic stability yields a gap in the spectrum of allowable labelled growth rates. As a further application, we obtain gap in the spectrum of allowable labelled growth rates in hereditary graph classes, with no a priori assumption of $\omega$-categoricity. We also establish a way to translate results about labelled growth rates of $\omega$-categorical structures into combinatorial statements about sets with weak finiteness properties in the absence of the axiom of choice, and derive several results from this translation.