Approximate Graph Colouring and Crystals

We show that approximate graph colouring is not solved by any level of the affine integer programming (AIP) hierarchy. To establish the result, we translate the problem of exhibiting a graph fooling a level of the AIP hierarchy into the problem of constructing a highly symmetric crystal tensor. In order to prove the existence of crystals in arbitrary dimension, we provide a combinatorial characterisation for realisable systems of tensors; i.e., sets of low-dimensional tensors that can be realised as the projections of a single high-dimensional tensor.