Operators in the Hilbert Space: the Ramsey Approach

Ramsey theory is applied to the analysis of operators acting on the functions belonging to the L^2 Hilbert space. The operators form the vertices of the bi-colored graph. If the operators commute, they are connected by a red link; if the operators do not commute they are connected with a green link. Thus, the complete, bi-colored graph emerges and the Ramsey theory becomes applicable. If the graph contains six vertices/operators, at least one monochromatic triangle will necessarily appear in the graph. Thus, the triad of operators forming the read triangle possesses the common set of eigenfunctions. The extension of introduced approach to infinite sets of operators is addressed. Applications of the introduced approach to problems of classical and quantum mechanics are suggested.