## A limit law of almost $$l$$-partite graphs.(English)Zbl 1325.03033

This article concludes that for any integers $$1 \leq s_1 \leq \cdots \leq s_l$$, if $${\mathcal K} = {\mathcal K}_{1, s_1, \ldots, s_l}$$ is the complete $$(l + 1)$$-partite graph whose parts consist of $$1$$, $$s_1,\dots,s_l$$ vertices, respectively, and if $$\mathbf {Forb}(\mathcal K)$$ is the set of graphs not having $$\mathcal K$$ as a subgraph, then $$\mathbf {Forb}(\mathcal K)$$ has a (labelled) first order limit law.
In order to establish this result, the article further develops the theory of almost $$l$$-partite graphs. Given positive integers $$l$$ and $$d$$, let $$\mathbf P_n (l, d)$$ be the set of $$n$$-vertex graphs that can be partitioned into $$l$$ parts so that each vertex is adjacent to at most $$d$$ vertices in its own part: thus $$\mathbf P_n(l, 0)$$ is the set of $$l$$-partite graphs. The primary result in this paper is that for each positive integer $$l$$, $$\mathbf P(l, 1)$$ admits a (labelled) first order zero-one law, while for each positive integer $$d > 1$$, $$\mathbf P(l, d)$$ admits a first order limit law but also first order properties whose limit is neither $$0$$ nor $$1$$.
The proofs are elementary but technically intricate, and the article is accessible to a reader willing to invest the effort.

### MSC:

 03C13 Model theory of finite structures 05C80 Random graphs (graph-theoretic aspects)
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