Summary: A class of graphs is \(\chi\)-bounded if there is a function \(f\) such that \(\chi(G) \leq f(\omega(G))\) for every induced subgraph \(G\) of every graph in the class, where \(\chi, \omega\) denote the chromatic number and clique number of \(G\) respectively. A. Gyárfás [Zastosow. Mat. 19, No. 3–4, 413–441 (1987; Zbl 0718.05041)] conjectured that for every \(c\), if \(\mathcal{C}\) is a class of graphs such that \(\chi(G) \leq \omega(G) + c\) for every induced subgraph \(G\) of every graph in the class, then the class of complements of members of \(\mathcal{C}\) is \(\chi \)-bounded. We prove this conjecture. Indeed, more generally, a class of graphs is \(\chi \)-bounded if it has the property that no graph in the class has \(c + 1\) odd holes, pairwise disjoint and with no edges between them. The main tool is a lemma that if \(C\) is a shortest odd hole in a graph, and \(X\) is the set of vertices with at least five neighbours in \(V(C)\), then there is a three-vertex set that dominates \(X\).
For Part VI see [the authors, “Induced subgraphs of graphs with large chromatic number. VI. Banana trees”, Preprint, arXiv:1701.05597].