| Summary: | Abstract The constraints arising from anomaly cancellation are particular strong for chiral theories in six dimensions. We make progress towards a complete classification of 6D supergravities with minimal supersymmetry and non-abelian gauge group. First, we generalize a previously known infinite class of anomaly-free theories which has 𝑇 ≫ 9 to essentially any semi-simple gauge group and infinitely many choices for hypermultiplets. The construction relies on having many decoupled sectors all selected from a list of four simple theories which we identify. Second, we use ideas from graph theory to rephrase the task of finding anomaly-free theories as constructing cliques in a certain multigraph. A branch-and-bound type algorithm is described which can be used to explicitly construct, in a 𝑇-independent way, anomaly-free theories with an arbitrary number of simple factors in the gauge group. We implement these ideas to generate an ensemble of O $$ \mathcal{O} $$ (107) irreducible cliques from which anomaly-free theories may be easily built, and as a special case obtain a complete list of 19,847 consistent theories for 𝑇 = 0, for which the maximal gauge group rank is 24. Modulo U(1), SU(2) and SU(3) simple factors and the new infinite families, we give a complete characterization of anomaly-free theories and show that the bound 𝑇 ≤ 273 is sharp.
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