Ramsey Approach to Vector Fields

Ramsey approach to the vectors fields is introduced. Set of vectors defined on R^3, taken as the generators of the bi-colored complete graph, is introduced. Vectors are considered as the vertices of the graph. Following coloring procedure is suggested: vertices numbered i and j are linked with the red edge if their scalar product is non-negative; and they are linked with the green edge if their scalar product is negative. This procedure gives rise to the complete, bi-color graph. Graph generated by six vectors inevitably contains the monochromatic triangle: the Ramsey number Ring-like systems of generating vectors are addressed. Applications of the graphs emerging from the vector fields are discussed.