Minimum degree stability and locally colourable graphs

We tie together two natural but, a priori, different themes. As a starting point consider Erdős and Simonovits's classical edge stability for an $(r + 1)$-chromatic graph $H$. This says that any $n$-vertex $H$-free graph with $(1 − 1/r + o(1)){n \choose 2}$ edges is close to (within $o(n^2)$ edges of) $r$-partite. This is false if $1 − 1/r$ is replaced by any smaller constant. However, instead of insisting on many edges, what if we ask that the $n$-vertex graph has large minimum degree? This is the basic question of minimum degree stability: what constant $c$ guarantees that any $n$-vertex $H$-free graph with minimum degree greater than $cn$ is close to $r$-partite? $c$ depends not just on chromatic number of $H$ but also on its finer structure.

Somewhat surprisingly, answering the minimum degree stability question requires understanding locally colourable graphs -- graphs in which every neighbourhood has small chromatic number -- with large minimum degree. This is a natural local-to-global colouring question: if every neighbourhood is big and has small chromatic number must the whole graph have small chromatic number? The triangle-free case has a rich history. The more general case has some similarities but also striking differences.