Asymptotic enumeration of I3-free digraphs
نویسنده
چکیده
We prove that almost all digraphs not embedding an independent set of size 3 consist of two disjoint tournaments, and discuss connections with the theory of homogeneous simple structures. Our main result can be stated informally as saying that almost all finite directed graphs in which any three vertices span at least one directed edge consist of two tournaments with some directed edges between them. This is a directed-graphs version of the following theorem by Erdős, Kleitman, and Rothschild (Theorem 2 in [EKR]): Theorem 0.1. Let Tn be the number of labelled triangle-free graphs on a set of n vertices, and Sn be the number of labelled bipartite graphs on n vertices. Then Tn = Sn(1 + o( 1 n )). So the proportion of triangle-free graphs on n vertices that are not bipartite is negligible for large n. Recall that a sentence σ is almost surely true (respectively, almost surely false) if the fraction μn(σ) of structures with universe {0, . . . , n − 1} satisfying σ converges to 1 (0) as n approaches infinity. Fagin [Fag76] proved: Theorem 0.2. Fix a relational language L. For every first-order sentence σ over L, μn(σ) converges to 0 or to 1. Given an L-sentence τ with μn(τ) > 0 for all n, denote by μn(σ|τ) the conditional probability μn(σ|τ) = μn(σ ∧ τ)/μn(τ). These conditional probabilities need not converge, but for some special cases they do converge. Given a relational language L and appropriate τ , let Tas(L; τ) be the set of Lsentences σ with limn→∞ μn(σ|τ) = 1. We call this the almost sure theory of L. It follows from Gaifman’s [Gai64] and Fagin’s work that T is consistent and complete when τ is ∀x(x = x); Fagin proved in [Fag76] that T is also consistent and complete in the cases where L is the language {R} and τ expresses one of the following: 1. R is a graph relation, 2. R is a tournament predicate symbol.
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