Counting unlabeled structures

Abstract

In this note we prove that whenever l is an infinite class of finite labeled structures provided with one binary relation such that l is closed under isomorphisms and (induced) substructures and l is rich enough (in a quantitative sense) then almost all structures in l are rigid, i.e., have no nontrivial automorphism. Applying this result to well-known results for labeled graphs we derive, for example, that almost every unlabeled K l+1 -free graph is already l-colorable, and we obtain 0–1 laws for the classes of unlabeled K l+1 -free graphs. It is worth while to note that a special case of our result states that almost all partial orders are rigid. As a consequence of this and the Kleitman-Rothschiid theorem ( Trans. Amer. Math. Soc. 205 (1975), 205–220) we get an asymptotic formula for the number of unlabeled partial orders.