  The classical Andr\'{a}sfai-Erd\H{o}s-S\'{o}s theorem considers the chromatic
number of $K_{r + 1}$-free graphs with large minimum degree, and in the case $r
= 2$ says that any $n$-vertex triangle-free graph with minimum degree greater
than $2/5 \cdot n$ is bipartite. This began the study of the chromatic profile
of triangle-free graphs: for each $k$, what minimum degree guarantees that a
triangle-free graph is $k$-colourable? The profile has been extensively studied
and was finally determined by Brandt and Thomass\'{e}.
  Triangle-free graphs are exactly those in which each neighbourhood is
one-colourable. As a natural variant, \L uczak and Thomass\'{e} introduced the
notion of a locally bipartite graph in which each neighbourhood is
2-colourable. Here we study the chromatic profile of the family of graphs in
which every neighbourhood is $b$-colourable (locally $b$-partite graphs) as
well as the family where the common neighbourhood of every $a$-clique is
$b$-colourable. Our results include the chromatic thresholds of these families
as well as showing that every $n$-vertex locally $b$-partite graph with minimum
degree greater than $(1 - 1/(b + 1/7)) \cdot n$ is $(b + 1)$-colourable.
Understanding these locally colourable graphs is crucial for extending the
Andr\'{a}sfai-Erd\H{o}s-S\'{o}s theorem to non-complete graphs, which we
develop elsewhere.
