Jordan Curves: Ramsey Approach and Topology

We develop a topological-combinatorial framework applying classical Ramsey theory to systems of arcs connecting points on Jordan curves and their higher-dimensional analogues. A Jordan curve Λ partitions the plane into interior and exterior regions, enabling a canonical two-coloring of every arc connecting points on Λ according to whether its interior lies in Int(Λ) or Ext(Λ). Using this intrinsic coloring, we prove that any configuration of six points on Λ necessarily contains a monochromatic triangle, and that this property is invariant under all homeomorphisms of the plane. Extending the construction by including arcs lying on Λ itself yields a natural three-coloring, from which the classical value $R\left(3,3.3\right)=17$ guarantees the appearance of monochromatic triangles for sufficiently large point sets. For infinite point sets on Λ, the infinite Ramsey theorem ensures the existence of infinite monochromatic cliques, which we likewise show to be preserved under arbitrary topological deformations. The framework extends to Jordan surfaces and Jordan–Brouwer hypersurfaces in higher dimensions, where interior, exterior, and boundary regions again generate canonical colorings and Ramsey-type constraints. These results reveal a general principle: the separation properties of codimension-one topological boundaries induce universal combinatorial structures—such as monochromatic triangles and infinite monochromatic subsets—that are stable under continuous deformations. The approach offers new links between geometric topology, extremal combinatorics, and the analysis of constrained networks and interfaces.