Advanced Ramsey Dimensional Analysis

We propose the Ramsey approach for the dimensional analysis of physical systems, which is complementary to the seminal Buckingham theorem. Dimensionless constants describing the given physical systems are seen as the vertices of the graph, labeled at the “dimensions graph”. The vertices are connected with the aqua-colored link/edge if they contain at least one common for both of them dimensional physical value; the vertices are connected with the brown link/edge if they do not contain the common dimensional physical value for both of them. Thus, the bi-colored, complete, Ramsey graph emerges. The introduced relations between the vertices of the dimensions graph are non-transitive. According to the Ramsey theorem the mono-chromatic triangle will be necessarily present in the dimensions graph built of six vertices, whatever is the order of the vertices. The introduced Ramsey approach is extended for the dimensionless constants built of the fundamental physical constants. Physical interpretation of the Ramsey analysis of the dimensions graphs is suggested. The generalization of the introduced Ramsey scheme for multi-colored Ramsey graphs is addressed. Extension to the infinite sets of dimensionless constants is presented. Introduced dimensions graphs are invariant relatively to rotations of frames, but they are sensitive to the Galilean and Lorentz transformations.