Abstract
Ramsey theory is the study of structure that must exist in a system, most typically after it has been partitioned.
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References
H. Ardal, J. Maňuch, M. Rosenfeld, S. Shelah, and L. Stacho, The odd-distance plane graph. Discrete Comput. Geom., 42 (2009), 132–141.
M. Benda and M. Perles, Colorings of metric spaces. Geombinatorics, 9 (2000), 113–126.
K. B. Chilakamarri, Some problems arising from unit–distance graphs. Geombinatorics, 4(4) (1995), 104–109.
K. B. Chilakamarri, On the chromatic number of rational five-space. Aequationes Mathematicae, 39 (1990), 146–148.
D. A. Coulson, A 15-coloring of 3-space omitting distance one, Discrete Math. 256 (2002), 83–90.
D. A. Coulson and M. S. Payne, A dense distance 1 excluding set in ℝ 3. Aust. Math. Soc. Gazette, 34 (2007), 97–102.
H. Croft, Incidence incidents. Eureka, 30 (1967), 22–26.
P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus, Euclidean Ramsey Theory I, J. Combin. Theor. Ser. A 14 (1973), 341–363.
P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus, Euclidean Ramsey Theory III, Infinite and finite sets Colloq., Keszthely (1973), Vol. I, pp. 559–583. Colloq. Math. Soc. Janos Bolyai, Vol. 10, North-Holland, Amsterdam, 1975.
P. Frankl and V. Rödl, A partition property of simplices in Euclidean space. J. Am. Math. Soc., 3 (1990), 1–7.
L. L. Ivanov, An estimate for the chromatic number of the space ℝ 4. Russ. Math. Surv., 61 (2006), 984–986.
L. L. Ivanov, On the chromatic numbers of ℝ 2 and ℝ 3 with intervals of forbidden distances. Electron. Notes Discrete Math., 29 (2007), 159–162.
V. Jelínek, J. Kynčl, R. Stolař, and T. Valla, Monochromatic triangles in two-colored plane. arXiv.math/0701940v1.
I. Křiž, Permutation groups in Euclidean Ramsey theory. Proc. Am. Math. Soc. 112 (1991), 899–907.
I. Křiž, All trapezoids are Ramsey. Discrete Math. 108 (1992), 59–62.
K. Kuratowski, Sur le problème des courbes gauches en topologie. Fund. Math. 15 (1930), 271–283.
O. Nechushtan, On the space chromatic number. Discrete Math., 256 (2002), 499–507.
P. O’Donnell, Arbitrary girth, 4-chromatic unit distance graphs in the plane. I. Graph description. Geombinatorics 9 (2000), 145–152.
P. O’Donnell, Arbitrary girth, 4-chromatic unit distance graphs in the plane. II. Graph embedding. Geombinatorics 9 (2000), 180–193.
A. M. Raǐgorodoskiǐ and I. M. Shitova, Chromatic numbers of real and rational spaces with real or rational forbidden distances. Sbornik: Math., 199(4), 579–612.
D. E. Raiskii, Realizing of all distances in a decomposition of the space ℝ n into n+1 parts. Zametki, 9 (1970), 319–323.
L. Shader, All right triangles are Ramsey in E 2! J. Comb. Theor., Ser. A, 20 (1976), 385–389.
A. Soifer, Chromatic number of the plane & its relatives. I. The problem & its history. Geombinatorics 12 (2003), 131–148.
A. Soifer, Chromatic number of the plane & its relatives. II. Polychromatic number & 6-coloring. Geombinatorics 12 (2003), 191–216.
A. Soifer, Chromatic number of the plane & its relatives. III. Its future. Geombinatorics 13 (2003), 41–46.
A. Soifer, The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators. Springer, New York, 2009.
E. G. Straus Jr., A combinatorial theorem in group theory, Math. Comp. 29 (1975), 303–309.
B. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskunde 19 (1927), 212–216.
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Graham, R., Tressler, E. (2011). Open Problems in Euclidean Ramsey Theory. In: Soifer, A. (eds) Ramsey Theory. Progress in Mathematics, vol 285. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8092-3_7
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