Growth of Sobolev norms for the quintic NLS on ${\mathbb{T}}^2$

We study the quintic Non Linear Schr"odinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in the case of the cubic NLS. This requires an accurate combinatorial analysis.