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doi:10.1016/0097-3165(87)90061-6 
Copyright © 1987 Published by Elsevier Inc.
Counting unlabeled structures
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Hans Jürgen Prömel*
Received 18 April 1986.
Abstract
In this note we prove that whenever
is an infinite class of finite labeled structures provided with one binary relation such that
is closed under isomorphisms and (induced) substructures and
is rich enough (in a quantitative sense) then almost all structures in
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 Kl+1-free graph is already l-colorable, and we obtain 0–1 laws for the classes of unlabeled Kl+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.
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* Supported by Son derforschungsbereich 303 (DFG), Institut fur Ökonometnr und Operations Research, Universität Bonn, W. Germany.
