Abstract
We prove that for any partition of the plane into a closed set C and an open set O and for any configuration T of three points, there is a translated and rotated copy of T contained in C or in O.
Apart from that, we consider partitions of the plane into two sets whose common boundary is a union of piecewise linear curves. We show that for any such partition and any configuration T which is a vertex set of a non-equilateral triangle there is a copy of T contained in the interior of one of the two partition classes. Furthermore, we characterize the “polygonal” partitions that avoid copies of a given equilateral triple.
These results support a conjecture of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, which states that every two-coloring of the plane contains a monochromatic copy of any nonequilateral triple of points; on the other hand, we disprove a stronger conjecture by the same authors, by providing non-trivial examples of two-colorings that avoid a given equilateral triple.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Bóna and G. Tóth: A Ramsey-type problem on right-angled triangles in space, Discrete Math.250 (1996), 61–67.
P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus: Euclidean Ramsey Theorems I, Journal of Comb. Theory (A)14 (1973), 341–363.
P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus: Euclidean Ramsey Theorems II, Infinite and Finite Sets10 (1973), 529–557.
P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus: Euclidean Ramsey Theorems III, Infinite and Finite Sets10 (1973), 559–583.
P. Frankl and V. Rödl: A Partition Property of simplices in Euclidean space, Journal of the Amer. Math. Soc.3(1) (1990), 1–7.
R. L. Graham: Recent trends in Euclidean Ramsey theory, Discrete Math. 136 (1994), 119–127.
I. Kríž: Permutation groups in Euclidean Ramsey theory, Proc. Amer. Math. Soc. 112 (1991), 899–907.
I. Kríž: All trapezoids are Ramsey, Discrete Math. 108 (1992), 59–62.
J. Matousek and V. Rödl: On Ramsey Sets in Spheres, Journal of Comb. Theory (A)70 (1995), 30–44.
L. E. Shader: All right triangles are Ramsey in E2!, Journal of Comb. Theory (A)20 (1979), 385–389.
D. R. Woodall: Distances realized by sets covering the plane, Journal of Comb. Theory (A)14 (1973), 187–200.
Author information
Authors and Affiliations
Corresponding author
Additional information
ITI is supported by project 1M0021620808 of the Czech Ministry of Education.
KAM is supported by project MSM0021620838 of the Czech Ministry of Education.
Rights and permissions
About this article
Cite this article
Jelínek, V., Kynčl, J., Stolař, R. et al. Monochromatic triangles in two-colored plane. Combinatorica 29, 699–718 (2009). https://doi.org/10.1007/s00493-009-2291-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-009-2291-y