Strong Brandt-Thomass\'e Theorems

Solving a long standing conjecture of Erd{o}s and Simonovits, Brandt and Thomass\'e proved that the chromatic number of each triangle-free graph G such that δ(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∈\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.