Advanced Ramsey Dimensional Analysis

We propose a Ramsey approach to the dimensional analysis of physical systems, which complements the seminal Buckingham theorem. Dimensionless constants describing a given physical system are represented as vertices of a graph, referred to as a dimensions graph. Two vertices are connected by an aqua-colored edge if they share at least one common dimensional physical quantity and by a brown edge if they do not. In this way, a bi-colored complete Ramsey graph is obtained. The relations introduced between the vertices of the dimensions graph are non-transitive. According to the Ramsey theorem, a monochromatic triangle must necessarily appear in a dimensions graph constructed from six vertices, regardless of the order of the vertices. Mantel–Turán analysis is applied to study these graphs. The proposed Ramsey approach is extended to graphs constructed from fundamental physical constants. A physical interpretation of the Ramsey analysis of dimensions graphs is suggested. A generalization of the proposed Ramsey scheme to multi-colored Ramsey graphs is also discussed, along with an extension to infinite sets of dimensionless constants. The introduced dimensions graphs are invariant under rotations of reference frames, but they are sensitive to Galilean and Lorentz transformations. The coloring of the dimensions graph is independent of the chosen system of units. The number of vertices in a dimensions graph is relativistically invariant and independent of the system of units.