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Volumn , Issue , 2014, Pages 429-460

Coloring distance graphs and graphs of diameters

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EID: 84898002428     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-1-4614-0110-0_23     Document Type: Chapter
Times cited : (99)

References (123)
  • 1
    • 84866350828 scopus 로고    scopus 로고
    • Distance graphs with large chromatic numbers and small clique numbers
    • A.B. Kupavskii, A.M. Raigorodskii, Distance Graphs with Large Chromatic Numbers and Small Clique Numbers, Doklady Mathematics, 85 No. 3, 394-398 (2012)
    • (2012) Doklady Mathematics , vol.85 , Issue.3 , pp. 394-398
    • Kupavskii, A.B.1    Raigorodskii, A.M.2
  • 2
    • 85164405616 scopus 로고    scopus 로고
    • Distance graphs with large chromatic number and arbitrary girth
    • A.B. Kupavskii, Distance graphs with large chromatic number and arbitrary girth, Moscow J. of Comb. and Number Theory, 2(N2), 52-62 (2012)
    • (2012) Moscow J. of Comb. and Number Theory , vol.2 , Issue.N2 , pp. 52-62
    • Kupavskii, A.B.1
  • 5
    • 0030269011 scopus 로고    scopus 로고
    • The complete nontrivial-intersection theorem for systems of finite sets
    • R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets. J. Combin. Theor. A 76, 121-138 (1996)
    • (1996) J. Combin. Theor. A , vol.76 , pp. 121-138
    • Ahlswede, R.1    Khachatrian, L.H.2
  • 6
    • 21444442876 scopus 로고    scopus 로고
    • The complete intersection theorem for systems of finite sets
    • R. Ahlswede, L.H. Khachatrian, The complete intersection theorem for systems of finite sets. Eur. J. Combin. 18, 125-136 (1997)
    • (1997) Eur. J. Combin , vol.18 , pp. 125-136
    • Ahlswede, R.1    Khachatrian, L.H.2
  • 7
    • 44949282626 scopus 로고
    • Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson type intersection theorems
    • N. Alon, L. Babai, H. Suzuki, Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson type intersection theorems. J. Combin. Theor. A, 58, 165-180 (1991)
    • (1991) J. Combin. Theor. A , vol.58 , pp. 165-180
    • Alon, N.1    Babai, L.2    Suzuki, H.3
  • 8
    • 0003430191 scopus 로고    scopus 로고
    • 2nd edn. Wiley-Interscience Series in Discrete Mathematics and Optimization, New York
    • N. Alon, J. Spencer, The Probabilistic Method, 2nd edn. (Wiley-Interscience Series in Discrete Mathematics and Optimization, New York, 2000)
    • (2000) The Probabilistic Method
    • Alon, N.1    Spencer, J.2
  • 9
    • 0003536935 scopus 로고
    • Department of Computer Science, University of Chicago, Preliminary version 2, September
    • L. Babai, P. Frankl, in Linear Algebra Methods in Combinatorics, Part 1, Department of Computer Science, University of Chicago, Preliminary version 2, September 1992
    • (1992) Linear Algebra Methods in Combinatorics, Part 1
    • Babai, L.1    Frankl, P.2
  • 10
    • 4444329922 scopus 로고    scopus 로고
    • The difference between consecutive primes, II
    • R.C. Baker, G. Harman, J. Pintz, The difference between consecutive primes, II. Proc. Lond. Math. Soc. 83, 532-562 (2001)
    • (2001) Proc. Lond. Math. Soc , vol.83 , pp. 532-562
    • Baker, R.C.1    Harman, G.2    Pintz, J.3
  • 11
    • 0008334235 scopus 로고    scopus 로고
    • Colorings of metric spaces
    • M. Benda, M. Perles, Colorings of metric spaces. Geombinatorics 9, 113-126 (2000)
    • (2000) Geombinatorics , vol.9 , pp. 113-126
    • Benda, M.1    Perles, M.2
  • 12
    • 11944253901 scopus 로고    scopus 로고
    • 2nd edn. (Cambridge University Press, Cambridge
    • B. Bollobás, Random Graphs, 2nd edn. (Cambridge University Press, Cambridge, 2001)
    • (2001) Random Graphs
    • Bollobás, B.1
  • 14
    • 0001320722 scopus 로고
    • Drei Sätze über die n-dimensionale euklidische Sphäre
    • K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre. Fundamenta Math. 20, 177-190 (1933)
    • (1933) Fundamenta Math , vol.20 , pp. 177-190
    • Borsuk, K.1
  • 15
    • 0001574571 scopus 로고
    • On covering a set in Rd by balls of the same diameter
    • ed. by J. Lindenstrauss, V. Milman. Lecture Notes in Math., Springer-Verlag, Berlin
    • J. Bourgain, J. Lindenstrauss, On covering a set in Rd by balls of the same diameter, ed. by J. Lindenstrauss, V. Milman. Geometric Aspects of Functional Analysis, Lecture Notes in Math., vol. 1469 (Springer-Verlag, Berlin, 1991), pp. 138-144
    • (1991) Geometric Aspects of Functional Analysis , vol.1469 , pp. 138-144
    • Bourgain, J.1    Lindenstrauss, J.2
  • 17
    • 33846304301 scopus 로고
    • A colour problem for infinite graphs and a problem in the theory of relations
    • N.G. de Bruijn, P. Erdo″s, A colour problem for infinite graphs and a problem in the theory of relations. Proc. Koninkl. Nederl. Acad. Wet., Ser. A 54(N5), 371-373 (1951)
    • (1951) Proc. Koninkl. Nederl. Acad. Wet., Ser. A , vol.54 , Issue.N5 , pp. 371-373
    • De Bruijn, N.G.1    Erdos, P.2
  • 18
    • 1842625061 scopus 로고    scopus 로고
    • Finite euclidean ramsey theory
    • K. Cantwell, Finite Euclidean Ramsey theory. J. Combin. Theor. A 73(N2), 273-285 (1996)
    • (1996) J. Combin. Theor. A , vol.73 , Issue.N2 , pp. 273-285
    • Cantwell, K.1
  • 19
    • 0011471443 scopus 로고
    • The unit-distance graph problem: A brief survey and some new results
    • K.B. Chilakamarri, The unit-distance graph problem: A brief survey and some new results. Bull. Inst. Comb. Appl. 8, 39-60 (1993)
    • (1993) Bull. Inst. Comb. Appl , vol.8 , pp. 39-60
    • Chilakamarri, K.B.1
  • 20
    • 67651157586 scopus 로고    scopus 로고
    • On the chromatic number of real and rational spaces
    • J. Cibulka, On the chromatic number of real and rational spaces. Geombinatorics 18(N2), 53-66 (2008)
    • (2008) Geombinatorics , vol.18 , Issue.N2 , pp. 53-66
    • Cibulka, J.1
  • 22
    • 31244436597 scopus 로고    scopus 로고
    • A 15-coloring of 3-space omitting distance one
    • D. Coulson, A 15-coloring of 3-space omitting distance one. Discrete Math. 256, 83-90 (2002)
    • (2002) Discrete Math , vol.256 , pp. 83-90
    • Coulson, D.1
  • 23
    • 0039422435 scopus 로고
    • Incident incidents
    • H.T. Croft, Incident incidents. Eureka (Cambridge) 30, 22-26 (1967)
    • (1967) Eureka (Cambridge , vol.30 , pp. 22-26
    • Croft, H.T.1
  • 25
    • 38349028318 scopus 로고    scopus 로고
    • Some 4-chromatic unit-distance graphs without small cycles
    • P. O'Donnel, R. Hochberg, Some 4-chromatic unit-distance graphs without small cycles. Geombinatorics 5(N4), 137-142 (1996)
    • (1996) Geombinatorics , vol.5 , Issue.N4 , pp. 137-142
    • O'donnel, P.1    Hochberg, R.2
  • 26
    • 50249106662 scopus 로고    scopus 로고
    • Arbitrary girth, 4-chromatic unit distnace graphs in the plane. Graph description
    • P. O'Donnell, Arbitrary girth, 4-chromatic unit distnace graphs in the plane. Graph description. Geombinatorics 9, 145-152 (2000)
    • (2000) Geombinatorics , vol.9 , pp. 145-152
    • O'donnell, P.1
  • 27
    • 50249106662 scopus 로고    scopus 로고
    • Arbitrary girth, 4-chromatic unit distance graphs in the plane Graph embedding
    • P. O'Donnell, Arbitrary girth, 4-chromatic unit distance graphs in the plane. Graph embedding. Geombinatorics 9, 180-193 (2000)
    • (2000) Geombinatorics , vol.9 , pp. 180-193
    • O'donnell, P.1
  • 28
    • 84963015763 scopus 로고
    • Covering a three-dimensional set with sets of smaller diameter
    • H.G. Eggleston, Covering a three-dimensional set with sets of smaller diameter. J. Lond. Math. Soc. 30, 11-24 (1955)
    • (1955) J. Lond. Math. Soc , vol.30 , pp. 11-24
    • Eggleston, H.G.1
  • 30
    • 0000314230 scopus 로고
    • On sets of distances of n points
    • P. Erdo″s, On sets of distances of n points. Am. Math. Monthly 53, 248-250 (1946)
    • (1946) Am. Math. Monthly , vol.53 , pp. 248-250
    • Erdos, P.1
  • 32
    • 84960611591 scopus 로고
    • Intersection theorems for systems of finite sets
    • P. Erdo″s, C. Ko, R. Rado, Intersection theorems for systems of finite sets. J. Math. Oxford, Sec. 12(48) (1961)
    • (1961) J. Math. Oxford, Sec , vol.12 , pp. 48
    • Erdos, P.1    Ko, C.2    Rado, R.3
  • 36
    • 0002950863 scopus 로고
    • Probabilities in finite models
    • R. Fagin, Probabilities in finite models. J. Symbolic Logic 41, 50-58 (1976)
    • (1976) J. Symbolic Logic , vol.41 , pp. 50-58
    • Fagin, R.1
  • 37
    • 0043131507 scopus 로고
    • The realization of distances in measurable subsets covering Rn
    • K.J. Falconer, The realization of distances in measurable subsets covering Rn. J. Combin. Theor. A 31, 187-189 (1981)
    • (1981) J. Combin. Theor. A , vol.31 , pp. 187-189
    • Falconer, K.J.1
  • 38
    • 77956977317 scopus 로고    scopus 로고
    • Fourier analysis, linear programming, and densities of distance avoiding sets in Rn
    • F.M. de Oliveira Filho, F. Vallentin, Fourier analysis, linear programming, and densities of distance avoiding sets in Rn. J. Eur. Math. Soc. 12, 1417-1428 (2010)
    • (2010) J. Eur. Math. Soc , vol.12 , pp. 1417-1428
    • Filho Oliveira De, F.M.1    Vallentin, F.2
  • 39
    • 77958533677 scopus 로고    scopus 로고
    • Covering planar sets
    • V.P. Filimonov, Covering planar sets. Sb. Math. 201(N8), 1217-1248 (2010)
    • (2010) Sb. Math , vol.201 , Issue.N8 , pp. 1217-1248
    • Filimonov, V.P.1
  • 40
    • 67650093805 scopus 로고    scopus 로고
    • Division codes and an area theorem
    • V.P. Filimonov, Division codes and an area theorem. Dokl. Math. 79(N3), 345-348 (2009)
    • (2009) Dokl. Math , vol.79 , Issue.N3 , pp. 345-348
    • Filimonov, V.P.1
  • 41
    • 77954872562 scopus 로고    scopus 로고
    • Small deviation theorem for division codes
    • V.P. Filimonov, Small deviation theorem for division codes. Dokl. Math. 81(N2), 278-281 (2010)
    • (2010) Dokl. Math , vol.81 , Issue.N2 , pp. 278-281
    • Filimonov, V.P.1
  • 42
    • 84967800049 scopus 로고
    • Forbidden intersections
    • P. Frankl, V. Rödl, Forbidden intersections. Trans. Am. Math. Soc. 300(N1), 259-286 (1987)
    • (1987) Trans. Am. Math. Soc , vol.300 , Issue.N1 , pp. 259-286
    • Frankl, P.1    Rödl, V.2
  • 43
    • 51249185135 scopus 로고
    • Intersection theorems with geometric consequences
    • P. Frankl, R. Wilson, Intersection theorems with geometric consequences . Combinatorica 1, 357-368 (1981)
    • (1981) Combinatorica , vol.1 , pp. 357-368
    • Frankl, P.1    Wilson, R.2
  • 44
    • 34250470313 scopus 로고
    • Range and degree of realizability of formulas in the restricted predicate calculus
    • Yu.V. Glebskii, D.I. Kogan, M.I. Legon'kii, V.A. Talanov, Range and degree of realizability of formulas in the restricted predicate calculus. Cybernetics 5, 142-154 (1969)
    • (1969) Cybernetics , vol.5 , pp. 142-154
    • Glebskii, Yu.V.1    Kogan, D.I.2    Legon'kii, M.I.3    Talanov, V.A.4
  • 45
    • 70350164286 scopus 로고    scopus 로고
    • Estimating the chromatic numbers of Euclidean spaces by methods of convex minimization
    • E.S. Gorskaya, I.M. Mitricheva, V.Y. Protasov, A.M. Raigorodskii, Estimating the chromatic numbers of Euclidean spaces by methods of convex minimization. Sb. Math. 200(N6), 783-801 (2009)
    • (2009) Sb. Math , vol.200 , Issue.N6 , pp. 783-801
    • Gorskaya, E.S.1    Mitricheva, I.M.2    Protasov, V.Y.3    Raigorodskii, A.M.4
  • 47
    • 77949303516 scopus 로고    scopus 로고
    • On the independence numbers of distance graphs with vertices at {-1,0,1} n: Estimates, conjectures, and applications to the Borsuk and Nelson-Erdo″s-Hadwiger problems
    • A.E. Guterman, V.K. Lyubimov, A.M. Raigorodskii, A.S. Usachev, On the independence numbers of distance graphs with vertices at {-1,0,1}n: estimates, conjectures, and applications to the Borsuk and Nelson-Erdo″s-Hadwiger problems. J. Math. Sci. 165(N6), 689-709 (2010)
    • (2010) J. Math. Sci , vol.165 , Issue.N6 , pp. 689-709
    • Guterman, A.E.1    Lyubimov, V.K.2    Raigorodskii, A.M.3    Usachev, A.S.4
  • 48
    • 0000079242 scopus 로고
    • Ein überdeckungssatz fur den euklidischen raum
    • H. Hadwiger, Ein Überdeckungssatz fur den Euklidischen Raum. Portugaliae Math. 4, 140-144 (1944)
    • (1944) Portugaliae Math , vol.4 , pp. 140-144
    • Hadwiger, H.1
  • 49
    • 51249195232 scopus 로고
    • Überdeckung einer menge durch mengen kleineren durchmessers
    • Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers. Comm. Math. Helv. 19, 72-73 (1946/47
    • H. Hadwiger, Überdeckung einer Menge durch Mengen kleineren Durchmessers. Comm. Math. Helv. 18, 73-75 (1945/46); Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers. Comm. Math. Helv. 19, 72-73 (1946/47)
    • (1945) Comm. Math. Helv , vol.18 , pp. 73-75
    • Hadwiger, H.1
  • 51
    • 84928927885 scopus 로고    scopus 로고
    • On the chromatic numbers of r2 and r3 with intervals of forbidden distances
    • [in Russian]
    • L.L. Ivanov, On the chromatic numbers of R2 and R3 with intervals of forbidden distances. Vestnik of the Russian University of People's Friendship (N1), 14-29 (2011) [in Russian]
    • (2011) Vestnik of the Russian University of People's Friendship N1 , pp. 14-29
    • Ivanov, L.L.1
  • 52
    • 33947227321 scopus 로고    scopus 로고
    • An estimate of the chromatic number of the space R4
    • L.L. Ivanov, An estimate of the chromatic number of the space R4. Russ. Math. Surv. 61(N5), 984-986 (2006)
    • (2006) Russ. Math. Surv , vol.61 , Issue.N5 , pp. 984-986
    • Ivanov, L.L.1
  • 53
    • 49049136095 scopus 로고
    • Coloring abelian groups
    • P.D. Johnson, Coloring abelian groups. Discrete Math. 40, 219-223 (1982)
    • (1982) Discrete Math , vol.40 , pp. 219-223
    • Johnson, P.D.1
  • 54
    • 84887885299 scopus 로고    scopus 로고
    • Coloring the rational points to forbid the distance 1-A tentative history and compendium
    • P.D. Johnson, Coloring the rational points to forbid the distance 1-a tentative history and compendium. Geombinatorics 16, 209-218 (2006)
    • (2006) Geombinatorics , vol.16 , pp. 209-218
    • Johnson, P.D.1
  • 55
    • 84942225364 scopus 로고
    • Über die kleinste Kugel, die eine raümliche Figur einschliesst
    • H.W.E. Jung, Über die kleinste Kugel, die eine raümliche Figur einschliesst. J. Reine und Angew. Math. 123, 241-257 (1901)
    • (1901) J. Reine und Angew. Math , vol.123 , pp. 241-257
    • Jung, H.W.E.1
  • 56
    • 84967774552 scopus 로고
    • A counterexample to Borsuk's conjecture
    • J. Kahn, G. Kalai, A counterexample to Borsuk's conjecture. Bull. (new series) AMS 29(N1), 60-62 (1993)
    • (1993) Bull. (New Series AMS , vol.29 , Issue.N1 , pp. 60-62
    • Kahn, J.1    Kalai, G.2
  • 57
    • 2442589431 scopus 로고    scopus 로고
    • Distance graphs on Zn with l1-norm
    • J.-H. Kang, Z. Füredi, Distance graphs on Zn with l1-norm. Theor. Comp. Sci. 319(N1-3), 357-366 (2004)
    • (2004) Theor. Comp. Sci , vol.319 , Issue.N1-3 , pp. 357-366
    • Kang, J.-H.1    Füredi, Z.2
  • 58
    • 0035651827 scopus 로고    scopus 로고
    • Chromatic numbers of Cayley graphs on Z and recurrence
    • Y. Katznelson, Chromatic numbers of Cayley graphs on Z and recurrence. Combinatorica 21, 211-219 (2001)
    • (2001) Combinatorica , vol.21 , pp. 211-219
    • Katznelson, Y.1
  • 60
    • 84975965997 scopus 로고
    • An approximative theorem for Borsuk's conjecture
    • R. Knast, An approximative theorem for Borsuk's conjecture. Proc. Cambridge Phil. Soc. (N1), 75-76 (1974)
    • (1974) Proc. Cambridge Phil. Soc , Issue.N1 , pp. 75-76
    • Knast, R.1
  • 61
    • 84862120315 scopus 로고    scopus 로고
    • On coloring spheres embedded into Rn
    • A.B. Kupavskiy, On coloring spheres embedded into Rn. Sb. Math. 202(N6), 83-110 (2011)
    • (2011) Sb. Math , vol.202 , Issue.N6 , pp. 83-110
    • Kupavskiy, A.B.1
  • 62
    • 80052765569 scopus 로고    scopus 로고
    • On lifting of estimation of chromatic number of Rn in higher dimension
    • A.B. Kupavskiy, On lifting of estimation of chromatic number of Rn in higher dimension. Dokl. Math. 429(N3), 305-308 (2009)
    • (2009) Dokl. Math , vol.429 , Issue.N3 , pp. 305-308
    • Kupavskiy, A.B.1
  • 63
    • 78650782791 scopus 로고    scopus 로고
    • On the chromatic number of Rn with an arbitrary norm
    • A.B. Kupavskii, On the chromatic number of Rn with an arbitrary norm. Discrete Math. 311, 437-440 (2011)
    • (2011) Discrete Math , vol.311 , pp. 437-440
    • Kupavskii, A.B.1
  • 64
    • 79952902709 scopus 로고    scopus 로고
    • The chromatic number of Rn with a set of forbidden distances
    • A.B. Kupavskii, The chromatic number of Rn with a set of forbidden distances. Dokl. Math. 82(N3), 963-966 (2010)
    • (2010) Dokl. Math , vol.82 , Issue.N3 , pp. 963-966
    • Kupavskii, A.B.1
  • 65
    • 73249148833 scopus 로고    scopus 로고
    • On the chromatic number of R9
    • A.B. Kupavskiy, A.M. Raigorodskii, On the chromatic number of R9. J.Math. Sci. 163(N6), 720-731 (2009)
    • (2009) J.Math. Sci , vol.163 , Issue.N6 , pp. 720-731
    • Kupavskiy, A.B.1    Raigorodskii, A.M.2
  • 66
    • 84928934400 scopus 로고    scopus 로고
    • Counterexamples to Borsuk's conjecture on spheres of small radii, submitted to Moscow
    • It is accepted and will be published in 2 available at arXiv: 1010.0383
    • A.B. Kupavskii, A.M. Raigorodskii, Counterexamples to Borsuk's conjecture on spheres of small radii, submitted to Moscow J. Comb. and Number Theory. It is accepted and will be published in 2(3) (2012). available at arXiv: 1010.0383
    • (2012) J. Comb. and Number Theory , vol.3
    • Kupavskii, A.B.1    Raigorodskii, A.M.2
  • 67
    • 51249183723 scopus 로고
    • A note on the realization of distances within sets in Euclidean space
    • D.G. Larman, A note on the realization of distances within sets in Euclidean space. Comment. Math. Helvet. 53, 529-535 (1978)
    • (1978) Comment. Math. Helvet , vol.53 , pp. 529-535
    • Larman, D.G.1
  • 68
    • 84976175804 scopus 로고
    • The realization of distances within sets in Euclidean space
    • D.G. Larman, C.A. Rogers, The realization of distances within sets in Euclidean space. Mathematika 19, 1-24 (1972)
    • (1972) Mathematika , vol.19 , pp. 1-24
    • Larman, D.G.1    Rogers, C.A.2
  • 69
    • 0012792395 scopus 로고
    • An estimate concerning Borsuk's partition problem
    • M. Lassak, An estimate concerning Borsuk's partition problem. Bull. Acad. Polon. Sci. Ser. Math. 30, 449-451 (1982)
    • (1982) Bull. Acad. Polon. Sci. Ser. Math , vol.30 , pp. 449-451
    • Lassak, M.1
  • 70
    • 25444474350 scopus 로고
    • Zerlegung ebener bereiche in konvexe zellen von möglichst kleinern durchmesser
    • H. Lenz, Zerlegung ebener Bereiche in konvexe Zellen von möglichst kleinern Durchmesser. Jahresbericht d. DMV Bd. 58, 87-97 (1956)
    • (1956) Jahresbericht D. DMV , vol.58 , pp. 87-97
    • Lenz, H.1
  • 71
    • 25444498476 scopus 로고
    • Über die bedeckung ebener punktmengen durch solche kleineren durchmessers
    • H. Lenz, Über die Bedeckung ebener Punktmengen durch solche kleineren Durchmessers. Arch. Math. VII, 34-40 (1956)
    • (1956) Arch. Math , vol.7 , pp. 34-40
    • Lenz, H.1
  • 72
    • 11744379524 scopus 로고
    • Self-dual polytopes and the chromatic number of distance graphs on the sphere
    • L. Lovász, Self-dual polytopes and the chromatic number of distance graphs on the sphere. Acta Sci. Math. 45, 317-323 (1983)
    • (1983) Acta Sci. Math , vol.45 , pp. 317-323
    • Lovász, L.1
  • 73
    • 69549088481 scopus 로고    scopus 로고
    • On lower bounds for the independence numbers of some distance graphs with vertices at {-1,0,1} n
    • V.K. Lyubimov, A.M. Raigorodskii, On lower bounds for the independence numbers of some distance graphs with vertices at {-1,0,1}n. Dokl. Math. 80(N1), 547-549 (2009)
    • (2009) Dokl. Math , vol.80 , Issue.N1 , pp. 547-549
    • Lyubimov, V.K.1    Raigorodskii, A.M.2
  • 75
    • 84887932299 scopus 로고    scopus 로고
    • Hunting unit-distance graphs in rational n-spaces
    • M. Mann, Hunting unit-distance graphs in rational n-spaces. Geombinatorics 13(N2), 49-53 (2003)
    • (2003) Geombinatorics , vol.13 , Issue.N2 , pp. 49-53
    • Mann, M.1
  • 77
    • 34547309583 scopus 로고    scopus 로고
    • On colouring the space Rn with several forbidden distances
    • N.G. Moshchevitin, A.M. Raigorodskii, On colouring the space Rn with several forbidden distances. Math. Notes 81(N5), 656-664 (2007)
    • (2007) Math. Notes , vol.81 , Issue.N5 , pp. 656-664
    • Moshchevitin, N.G.1    Raigorodskii, A.M.2
  • 78
    • 79952432802 scopus 로고    scopus 로고
    • New lower bounds for the independence numbers of distance graphs with vertices at {-1,0,1} n
    • V.F. Moskva, A.M. Raigorodskii, New lower bounds for the independence numbers of distance graphs with vertices at {-1,0,1}n. Math. Notes 89(N2), 307-308 (2011)
    • (2011) Math. Notes , vol.89 , Issue.N2 , pp. 307-308
    • Moskva, V.F.1    Raigorodskii, A.M.2
  • 79
    • 31244431950 scopus 로고    scopus 로고
    • Note on the space chromatic number
    • O. Nechushtan, Note on the space chromatic number. Discrete Math. 256, 499-507 (2002)
    • (2002) Discrete Math , vol.256 , pp. 499-507
    • Nechushtan, O.1
  • 80
    • 77952852566 scopus 로고    scopus 로고
    • Two Erdo″s problems on lacunary sequences: Chromatic numbers and Diophantine approximations
    • Y. Peres, W. Schlag, Two Erdo″s problems on lacunary sequences: chromatic numbers and Diophantine approximations, Bull. London Math. Soc. 42(2), 295-300 (2010)
    • (2010) Bull. London Math. Soc , vol.42 , Issue.2 , pp. 295-300
    • Peres, Y.1    Schlag, W.2
  • 82
    • 0041082751 scopus 로고    scopus 로고
    • Borsuk's problem and the chromatic numbers of metric spaces
    • A.M. Raigorodskii, Borsuk's problem and the chromatic numbers of metric spaces. Russ. Math. Surv. 56(N1), 103-139 (2001)
    • (2001) Russ. Math. Surv , vol.56 , Issue.N1 , pp. 103-139
    • Raigorodskii, A.M.1
  • 83
    • 66749129174 scopus 로고    scopus 로고
    • Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, [in Russian]
    • A.M. Raigorodskii, The Chromatic Numbers (Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, 2003) [in Russian]
    • (2003) The Chromatic Numbers
    • Raigorodskii, A.M.1
  • 84
    • 77949296376 scopus 로고    scopus 로고
    • Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, [in Russian]
    • A.M. Raigorodskii, Borsuk's Problem (Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, 2006) [in Russian]
    • (2006) Borsuk's Problem
    • Raigorodskii, A.M.1
  • 85
    • 54849423868 scopus 로고    scopus 로고
    • Around Borsuk's conjecture
    • A.M. Raigorodskii, Around Borsuk's conjecture. J. Math. Sci. 154(N4), 604-623 (2008)
    • (2008) J. Math. Sci , vol.154 , Issue.N4 , pp. 604-623
    • Raigorodskii, A.M.1
  • 86
    • 77950006360 scopus 로고    scopus 로고
    • Three lectures on the Borsuk partition problem
    • A.M. Raigorodskii, Three lectures on the Borsuk partition problem. Lond. Math. Soc. Lect. Note Ser. 347, 202-248 (2007)
    • (2007) Lond. Math. Soc. Lect. Note ser , vol.347 , pp. 202-248
    • Raigorodskii, A.M.1
  • 87
    • 4243180293 scopus 로고    scopus 로고
    • The Borsuk partition problem: The seventieth anniversary
    • A.M. Raigorodskii, The Borsuk partition problem: the seventieth anniversary. Math. Intel. 26(N3), 4-12 (2004)
    • (2004) Math. Intel , vol.26 , Issue.N3 , pp. 4-12
    • Raigorodskii, A.M.1
  • 88
    • 0034404299 scopus 로고    scopus 로고
    • On the chromatic number of a space
    • A.M. Raigorodskii, On the chromatic number of a space. Russ. Math. Surv. 55(N2), 351-352 (2000)
    • (2000) Russ. Math. Surv , vol.55 , Issue.N2 , pp. 351-352
    • Raigorodskii, A.M.1
  • 89
    • 36049022123 scopus 로고    scopus 로고
    • Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, [in Russian]
    • A.M. Raigorodskii, The Linear Algebra Method in Combinatorics (Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, 2007) [in Russian]
    • (2007) The Linear Algebra Method in Combinatorics
    • Raigorodskii, A.M.1
  • 90
    • 14544294314 scopus 로고    scopus 로고
    • On the chromatic number of a space with lq-norm
    • A.M. Raigorodskii, On the chromatic number of a space with lq-norm. Russ. Math. Surv. 59(N5), 973-975 (2004)
    • (2004) Russ. Math. Surv , vol.59 , Issue.N5 , pp. 973-975
    • Raigorodskii, A.M.1
  • 91
    • 77954899883 scopus 로고    scopus 로고
    • On the chromatic numbers of spheres in Euclidean spaces
    • A.M. Raigorodskii, On the chromatic numbers of spheres in Euclidean spaces. Dokl. Math. 81(N3), 379-382 (2010)
    • (2010) Dokl. Math , vol.81 , Issue.N3 , pp. 379-382
    • Raigorodskii, A.M.1
  • 92
    • 84862136768 scopus 로고    scopus 로고
    • On the chromatic numbers of spheres in Rn
    • A.M. Raigorodskii, On the chromatic numbers of spheres in Rn, Combinatorica, 32 (N1), 111-123 (2012)
    • (2012) Combinatorica , vol.32 , Issue.N1 , pp. 111-123
    • Raigorodskii, A.M.1
  • 93
    • 44449152154 scopus 로고    scopus 로고
    • On distance graphs with large chromatic number but not containing large simplexes
    • A.M. Raigorodskii, On distance graphs with large chromatic number but not containing large simplexes. Russ. Math. Surv. 62(N6), 1224-1225 (2007)
    • (2007) Russ. Math. Surv , vol.62 , Issue.N6 , pp. 1224-1225
    • Raigorodskii, A.M.1
  • 94
    • 18944367235 scopus 로고    scopus 로고
    • The Erdo″s-Hadwiger problem and the chromatic numbers of finite geometric graphs
    • A.M. Raigorodskii, The Erdo″s-Hadwiger problem and the chromatic numbers of finite geometric graphs. Sb. Math. 196(N1), 115-146 (2005)
    • (2005) Sb. Math , vol.196 , Issue.N1 , pp. 115-146
    • Raigorodskii, A.M.1
  • 95
    • 0033462933 scopus 로고    scopus 로고
    • On a bound in Borsuk's problem
    • A.M. Raigorodskii, On a bound in Borsuk's problem. Russ. Math. Surv. 54(N2), 453-454 (1999)
    • (1999) Russ. Math. Surv , vol.54 , Issue.N2 , pp. 453-454
    • Raigorodskii, A.M.1
  • 96
    • 78649952419 scopus 로고    scopus 로고
    • Counterexamples to Borsuk's conjecture on spheres of small radius
    • A.M. Raigorodskii, Counterexamples to Borsuk's conjecture on spheres of small radius. Dokl. Math. 82(N2), 719-721 (2010)
    • (2010) Dokl. Math , vol.82 , Issue.N2 , pp. 719-721
    • Raigorodskii, A.M.1
  • 97
    • 8644282834 scopus 로고    scopus 로고
    • On lower bounds for the Borsuk and Hadwiger numbers
    • A.M. Raigorodskii, On lower bounds for the Borsuk and Hadwiger numbers. Russ. Math. Surv. 59(N3), 585-586 (2004)
    • (2004) Russ. Math. Surv , vol.59 , Issue.N3 , pp. 585-586
    • Raigorodskii, A.M.1
  • 98
    • 33747865687 scopus 로고    scopus 로고
    • On the Borsuk and Erdo″s-Hadwiger numbers
    • A.M. Raigorodskii, On the Borsuk and Erdo″s-Hadwiger numbers. Math. Notes 79(N6), 854-863 (2006)
    • (2006) Math. Notes , vol.79 , Issue.N6 , pp. 854-863
    • Raigorodskii, A.M.1
  • 99
    • 79952909454 scopus 로고    scopus 로고
    • A lower bound for the chromatic number of the space Rn with k forbidden distances and metric l1
    • [in Russian]
    • A.M. Raigorodskii, M.I. Absalyamova, A lower bound for the chromatic number of the space Rn with k forbidden distances and metric l1. Chebyshev Sbornik 7(N4 (20)), 105-113 (2006) [in Russian]
    • (2006) Chebyshev Sbornik , vol.7 , Issue.20 N4 , pp. 105-113
    • Raigorodskii, A.M.1    Absalyamova, M.I.2
  • 100
    • 50849095141 scopus 로고    scopus 로고
    • On a series of questions concerning the Borsuk and the Nelson-Erdo″s-Hadwiger problems
    • A.M. Raigorodskii, M.M. Kityaev, On a series of questions concerning the Borsuk and the Nelson-Erdo″s-Hadwiger problems. Math. Notes 84(N2), 239-255 (2008)
    • (2008) Math. Notes , vol.84 , Issue.N2 , pp. 239-255
    • Raigorodskii, A.M.1    Kityaev, M.M.2
  • 101
    • 77950016199 scopus 로고    scopus 로고
    • Partition of three-dimensional sets into five parts of smaller diameter
    • A.M. Raigorodskii, A.B. Kupavskiy, Partition of three-dimensional sets into five parts of smaller diameter. Math. Notes 87(N2), 208-219 (2010)
    • (2010) Math. Notes , vol.87 , Issue.N2 , pp. 208-219
    • Raigorodskii, A.M.1    Kupavskiy, A.B.2
  • 103
    • 84928923924 scopus 로고    scopus 로고
    • ed. by S.D. Adhikari, B. Ramakrishnan, Harish-Chandra Research Institute. Number Theory and Applications: Proceedings of the International Conferences on Number Theory and Cryptography (A publication of It's located at Gurgaon, India
    • A.M. Raigorodskii, O.I. Rubanov, in On the Clique and the Chromatic Numbers of High-Dimensional Distance Graphs, ed. by S.D. Adhikari, B. Ramakrishnan, Harish-Chandra Research Institute. Number Theory and Applications: Proceedings of the International Conferences on Number Theory and Cryptography (A publication of It's located at Gurgaon, India, 2009), pp. 149-157
    • (2009) On the Clique and the Chromatic Numbers of High-Dimensional Distance Graphs , pp. 149-157
    • Raigorodskii, A.M.1    Rubanov, O.I.2
  • 104
    • 77954020616 scopus 로고    scopus 로고
    • On distance graphs with large chromatic number and without large cliques
    • A.M. Raigorodskii, O.I. Rubanov, On distance graphs with large chromatic number and without large cliques. Math. Notes 87(N3), 417-428 (2010)
    • (2010) Math. Notes , vol.87 , Issue.N3 , pp. 417-428
    • Raigorodskii, A.M.1    Rubanov, O.I.2
  • 105
    • 48049122798 scopus 로고    scopus 로고
    • On the chromatic numbers of real and rational spaces with several real or rational forbidden distances
    • A.M. Raigorodskii, I.M. Shitova, On the chromatic numbers of real and rational spaces with several real or rational forbidden distances. Sb. Math. 199(N4), 579-612 (2008)
    • (2008) Sb. Math , vol.199 , Issue.N4 , pp. 579-612
    • Raigorodskii, A.M.1    Shitova, I.M.2
  • 106
    • 43749124413 scopus 로고    scopus 로고
    • On the chromatic number of the Euclidean space and on Borsuk's problem
    • A.M. Raigorodskii, I.M. Shitova, On the chromatic number of the Euclidean space and on Borsuk's problem. Math. Notes 83(N4), 579-582 (2008)
    • (2008) Math. Notes , vol.83 , Issue.N4 , pp. 579-582
    • Raigorodskii, A.M.1    Shitova, I.M.2
  • 107
    • 0043032713 scopus 로고    scopus 로고
    • New sets with large Borsuk numbers
    • C. Richter, A. Hinrichs, New sets with large Borsuk numbers. Discrete Math. 270, 136-146 (2003)
    • (2003) Discrete Math , vol.270 , pp. 136-146
    • Richter, C.1    Hinrichs, A.2
  • 108
    • 84971146089 scopus 로고
    • Covering a sphere with spheres
    • C.A. Rogers, Covering a sphere with spheres. Mathematika 10, 157-164 (1963)
    • (1963) Mathematika , vol.10 , pp. 157-164
    • Rogers, C.A.1
  • 109
    • 84976102475 scopus 로고
    • Symmetrical sets of constant width and their partitions
    • C.A. Rogers, Symmetrical sets of constant width and their partitions. Mathematika 18, 105-111 (1971)
    • (1971) Mathematika , vol.18 , pp. 105-111
    • Rogers, C.A.1
  • 110
    • 38349050515 scopus 로고    scopus 로고
    • Chromatic numbers of 3-dimensional distance graphs containing no tetrahedra
    • O.I. Rubanov, Chromatic numbers of 3-dimensional distance graphs containing no tetrahedra. Math. Notes 82(N5-6), 718-721 (2007)
    • (2007) Math. Notes , vol.82 , Issue.N5-6 , pp. 718-721
    • Rubanov, O.I.1
  • 111
    • 84971761881 scopus 로고
    • Illuminating sets of constant width
    • O. Schramm, Illuminating sets of constant width. Mathematika 35, 180-189 (1988)
    • (1988) Mathematika , vol.35 , pp. 180-189
    • Schramm, O.1
  • 112
    • 33750993376 scopus 로고
    • On a problem of Erdo″s concerning 3-colouring of the unit sphere
    • G.J. Simmons, On a problem of Erdo″s concerning 3-colouring of the unit sphere. Discrete Math. 8, 81-84 (1974)
    • (1974) Discrete Math , vol.8 , pp. 81-84
    • Simmons, G.J.1
  • 113
    • 84974509716 scopus 로고
    • The chromatic number of the sphere
    • G.J. Simmons, The chromatic number of the sphere. J. Austral. Math. Soc. Ser. 21, 473-480 (1976)
    • (1976) J. Austral. Math. Soc. Ser. , vol.21 , pp. 473-480
    • Simmons, G.J.1
  • 115
    • 24544463963 scopus 로고    scopus 로고
    • Erdo″s on unit distances and the Szemerédi-Trotter theorems, Paul Erdo″s and his mathematics, bolyai series Budapest
    • L.A. Székely, Erdo″s on unit distances and the Szemerédi-Trotter theorems, Paul Erdo″s and his mathematics, bolyai series Budapest. J. Bolyai Math. Soc. 11, 649-666 (2002)
    • (2002) J. Bolyai Math. Soc , vol.11 , pp. 649-666
    • Székely, L.A.1
  • 116
    • 0041042329 scopus 로고
    • Bounds on the measurable chromatic number of Rn
    • L.A. Székely, N.C. Wormald, Bounds on the measurable chromatic number of Rn. Discrete Math. 75, 343-372 (1989)
    • (1989) Discrete Math , vol.75 , pp. 343-372
    • Székely, L.A.1    Wormald, N.C.2
  • 117
    • 0000510531 scopus 로고
    • Distances realized by sets covering the plane
    • D.R.Woodall, Distances realized by sets covering the plane. J. Combin. Theor. A 14, 187-200 (1973)
    • (1973) J. Combin. Theor. A , vol.14 , pp. 187-200
    • Woodall, D.R.1
  • 118
    • 84940300927 scopus 로고
    • A 4-chromatic graph with a special plane drawing
    • N. Wormald, A 4-chromatic graph with a special plane drawing. Aust. Mat. Soc. (Ser. A) 28, 1-8 (1979)
    • (1979) Aust. Mat. Soc. (Ser. A , vol.28 , pp. 1-8
    • Wormald, N.1
  • 119
    • 84867963117 scopus 로고    scopus 로고
    • On the connectivity of random distance graphs of a special type
    • A.R. Yarmukhametov, On the connectivity of random distance graphs of a special type. Chebyshev Sbornik 10(N1 (29)), 95-108 (2009)
    • (2009) Chebyshev Sbornik , vol.10 , Issue.29 N1 , pp. 95-108
    • Yarmukhametov, A.R.1
  • 120
    • 77950398246 scopus 로고    scopus 로고
    • The weak zero-one laws for the random distance graphs
    • M.E. Zhukovskii, The weak zero-one laws for the random distance graphs. Dokl. Math. 81(N1), 51-54 (2010)
    • (2010) Dokl. Math , vol.81 , Issue.N1 , pp. 51-54
    • Zhukovskii, M.E.1
  • 121
    • 79959324397 scopus 로고    scopus 로고
    • The weak zero-one law for the random distance graphs
    • M.E. Zhukovskii, The weak zero-one law for the random distance graphs. Theor. Prob. Appl. 55(N2), 344-350 (2010)
    • (2010) Theor. Prob. Appl , vol.55 , Issue.N2 , pp. 344-350
    • Zhukovskii, M.E.1
  • 122
    • 84855956911 scopus 로고    scopus 로고
    • The weak zero-one law for the random distance graphs
    • [in Russian]
    • M.E. Zhukovskii, The weak zero-one law for the random distance graphs. Vestnik Russ. Univ. People's Friendship 2(N1), 11-25 (2010) [in Russian]
    • (2010) Vestnik Russ. Univ. People's Friendship , vol.2 , Issue.N1 , pp. 11-25
    • Zhukovskii, M.E.1
  • 123
    • 0000009985 scopus 로고    scopus 로고
    • Coloring Hamming graphs, optimal binary codes, and the 0/1-Borsuk problem in low dimensions
    • G.M. Ziegler, Coloring Hamming graphs, optimal binary codes, and the 0/1-Borsuk problem in low dimensions. Lect. Notes Comput. Sci. 2122, 159-171 (2001)
    • (2001) Lect. Notes Comput. Sci , vol.2122 , pp. 159-171
    • Ziegler, G.M.1


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