A limit law of almost l-partite graphs

For integers $l \geq 2$, $d \geq 1$ we study (undirected) graphs with vertices $1, ..., n$ such that the vertices can be partitioned into $l$ parts such that every vertex has at most $d$ neighbours in its own part. The set of all such graphs is denoted $\mbP_n(l,d)$. We prove a labelled first-order limit law, i.e., for every first-order sentence $\varphi$, the proportion of graphs in $\mbP_n(l,d)$ that satisfy $\varphi$ converges as $n \to \infty$. By combining this result with a result of Hundack, Pr\"omel and Steger \cite{HPS} we also prove that if $1 \leq s_1 \leq ... \leq s_l$ are integers, then $\mb{Forb}(\mcK_{1, s_1, ..., s_l})$ has a labelled first-order limit law, where $\mb{Forb}(\mcK_{1, s_1, ..., s_l})$ denotes the set of all graphs with vertices $1, ..., n$, for some $n$, in which there is no subgraph isomorphic to the complete $(l+1)$-partite graph with parts of sizes $1, s_1, ..., s_l$. In the course of doing this we also prove that there exists a first-order formula $\xi$ (depending only on $l$ and $d$) such that the proportion of $\mcG \in \mbP_n(l,d)$ with the following property approaches 1 as $n \to \infty$: there is a unique partition of $\{1, ..., n\}$ into $l$ parts such that every vertex has at most $d$ neighbours in its own part, and this partition, viewed as an equivalence relation, is defined by $\xi$.