  Solving a long standing conjecture of Erd\H{o}s and Simonovits, Brandt and
Thomass\'e proved that the chromatic number of each triangle-free graph $G$
such that $\delta(G)>|V(G)|/3$ is at most four. In fact, they showed the much
stronger result that every maximal triangle-free graph $G$ satisfying this
minimum degree condition is a blow-up of either an Andr\'asfai or a Vega graph.
  Here we establish the same structural conclusion on $G$ under the weaker
assumption that for $m\in\{2, 3, 4\}$ every sequence of $3m$ vertices has a
subsequence of length $m+1$ with a common neighbour. In forthcoming work this
will be used to solve an old problem of Andr\'asfai in Ramsey-Tur\'an theory.
