Sum of squares lower bounds from symmetry and a good story

In this paper, we develop machinery for proving sum of squares lower bounds on symmetric problems based on the intuition that sum of squares has difficulty capturing integrality arguments, i.e. arguments that an expression must be an integer. Using this machinery, we prove a tight sum of squares lower bound for the following Turan type problem: Minimize the number of triangles in a graph $G$ with a fixed edge density. We also give an alternative proof of Grigoriev's sum of squares lower bound for the knapsack problem.