The Profile of relations

The {\it profile} of a relational structure $R$ is the function $\phi_R$ which counts for every integer $n$ the number of its $n$-element substructures up to an isomorphism. Many counting functions are profiles. Interesting examples come from permutation groups. Some salient facts about the behavior of the profile are presented. Techniques from ordered sets and combinatorics (notably the notion of well-quasi-order, the related notions of ordered algebras, Ramsey theorem) are illustrated. Ongoing resarch suggests to view the profile of a relational structure $R$ as the Hilbert function of some graded algebra associated with $R$. A hint at the solution of a conjecture of P.J.Cameron on the integrity of the ring of the orbit algebra is given. Recent progress made with Y.Boudabbous and N.Thi\'ery on the conjecture that the profile is a quasi-polynomial if its growth is polynomial (and the structure has a finite kernel) are presented.