Euclidean Ramsey Theory for Lines and Finite Configurations

A classical question in Euclidean Ramsey theory, posed by Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus, asks which finite configurations X⊂ E² must appear in blue under every red--blue coloring of the plane having no red pair of points distance one apart. For m ≥ 2, let ₘ denote the line of m points with consecutive points distance one apart. Tsaturian showed that ₅ is unavoidable; we improve this to ₇. Additionally, we show that in every three-coloring of the plane in which two colors contain no ₂, the third color must contain an ₃. To conclude our discussion on lines, we prove several related results on two-colorings of higher-dimensional Euclidean spaces where both colors avoid longer lines. Our methods also extend to other point configurations. In any two-coloring of the plane in which one color contains no ₂, we show that the other color must contain both a regular pentagon of side length one and a unit square together with its center. We also prove that every two-coloring of three-dimensional space contains either a monochromatic unit square or a monochromatic noncoplanar tetrahedron with four edges of length one and two of length √2. Our proofs combine Székely's averaging inequalities for distance-avoiding sets with the Fourier-analytic and linear-programming methods of de Oliveira Filho and Vallentin, together with a theorem on rotation-invariant measures from OpenAI's recent work on the Hadwiger-Nelson problem.