One of the most impactful tools for computer-assisted proofs in mathematics is due to Razborov, who introduced flag algebras in 2007 in order to study the limits of discrete objects in Extremal Combinatorics using concepts from First-Order Logic and Model Theory. They allow one to apply a Cauchy-Schwarz-type argument to classic problems in Turán and Ramsey theory by solving a concrete semidefinite programming formulation. In this work, we exploit symmetries in these formulations in order to reduce their size and extend the envelope of what is attainable using this technique. The resulting methods are based on combinatorial insights and a parameter-dependent notion of homomorphisms. To demonstrate their efficacy, we determine an precisely determine the 4-color Ramsey multiplicity of triangles