On metric Ramsey-type phenomena

Annals of Mathematics

Volumn 162, Issue 2, 2005, Pages 643-709

  Bartal, Yair ^a   Linial, Nathan ^b   Mendel, Manor ^c   Naor, Assaf ^d  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^b OPEN UNIVERSITY OF ISRAEL   (Israel)

^c Theory Group

^d MICROSOFT RESEARCH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 28944451802     PISSN: 0003486X     EISSN: 0003486X     Source Type: Journal    
DOI: 10.4007/annals.2005.162.643     Document Type: Review

References (52)

1. N. ALON, Eigenvalues and expanders, Combinatorial 6 (1986), 83-96.
2. K. BALL, Markov chains, Riesz transforms and Lipschitz maps, Geom. Fund. Anal. 2 (1992), 137-172.
3. Y. BARTAL, Probabilistic approximation of metric spaces and its algorithmic applications, in The 37th Annual Symposium on Foundations of Computer Science, pp. 184-193, 1996, IEEE Comput. Sci. Press, Los Alamitos, CA, 1996.
4. _,ACM New YorkOn approximating arbitrary metrics by tree metrics, in The 30th Annual ACM Symposium on Theory of Computing (STOC '98, Dallas, TX), 161-168, ACM, New York, 1999.
5. Y. BARTAL, B. BOLLOBÁS, and M. MENDEL, A Ramsey-type theorem for metric spaces and its applications for metrical task systems and related problems, in The 42nd Annual Symposium on Foundations of Computer Science, pp. 396-405 (Las Vegas, NV, 2001), IEEE Computer Soc., Los Alamitos, CA, 2001.
6. Y. BARTAL, N. LINIAL, M. MENDEL, and A. NAOR, Limitations to Fŕechet's metric embedding method, Israel J. Math., to appear.
7. _, Low dimensional embeddings of ultrametrics, European J. of Combinatorics 25 (2004), 87-92.
8. _, On some low distortion metric Ramsey problems, Discrete and Computational Geom. 33 (2005), 27-45.
9. _, On metric Ramsey-type phenomena (Conference version) 35th Annual ACM Symposium on Theory of Computing, pp. 463-472, ACM, New York, 2003.
10. Y. BARTAL and M. MENDEL, Multi-embedding and path-approximation of metric spaces, in Proc. First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 424-433 (Baltimore, MD, 2003), ACM, New York, 2003.
11. M. BEN-OR and N. LINIAL, Collective. Coin Flipping (S. Micali, ed.), Academic Press, New York, pp. 91-115, 1989.
12. Y. BENYAMINI and J. LINDENSTRAUSS, Geometric Nonlinear Functional Analysis. Vol. 1, Amer. Math. Soc. Colloq. Publ. 48, A. M. S., Providence, RI, 2000.
13. N. BIGGS, Algebraic Graph Theory, Cambridge Tracts in Math. 67, Cambridge Univ. Press, Cambridge, 1974.
14. A. BLUM, H. KARLOFF, Y. RABANI, and M. SAKS, A decomposition theorem for task systems and bounds for randomized server problems, SIAM J. Comput. 30 (2000), 1624-1661 (electronic).
15. J. BOURGAIN, On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J. Math. 52 (1985), 46-52.
16. _, The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math. 56 (1986), 222-230.
17. J. BOURGAIN, T. FIGIEL, and V. MILMAN, On Hilbertian subsets of finite metric spaces, Israel J. Math. 55 (1986), 147-152.
18. J. BOURGAIN, V. MILMAN, and H. WOLFSON, On types of metric spaces, Transl. Amer. Math. Soc. 294 (1986), 295-317.
19. 1, in Proc. 44th Annual IEEE Conference on Foundations of Computer Science, pp. 514-523, ACM, New York, 2003.
20. F. R. K. CHUNG, Diameters and eigenvalues, J. Amer. Math. Soc. 2 (1989), 187-196.
21. P. DELSARTE, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. 10 (1973), vi+97.
22. M. M. DEZA and M. LAURENT, Geometry of Cuts and Metrics, Springer-Verlag, New York, 1997.
23. J. DIESTEL, H. JARCHOW, and A. TONGE, Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, 1995.
24. A. DVORETZKY, Some results on convex bodies and Banach spaces, in Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 123-160, Jerusalem Academic Press, Jerusalem, 1961.
25. p,-spaces, Arkiv Mat. 8 (1969), 103-105.
26. U. FEIGE, Approximating the bandwidth via volume respecting embeddings, J. Comput. System Sci. 60 (2000), 510-539, 30th Annual ACM Symposium on Theory of Computing (Dallas, TX, 1998).
27. T. FIGIEL, J. LINDENSTRAUSS, and V. D. MILMAN, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53-94.
28. J. FAKCHAROENPHOL, S. RAO, and K. TALWAR, A tight bound on approximating arbitrary metrics by tree metrics, in Proc. 35th Annual ACM Symposium on Theory of Computing (2003), 448-455.
29. R. L. GRAHAM, B. L. ROTHSCHILD, and J. H. SPENCER, Ramsey Theory, second edition, John Wiley & Sons Inc., New York, 1990.
30. P. INDYK, Algorithmic applications of low-distortion geometric embeddings, in 42nd Annual Symposium on Foundations of Computer Science, pp. 10-33 (Las Vegas, NV, 2001), IEEE Computer Soc., Los Alamitos, CA, 2001.
31. W. B. JOHNSON and J. LINDENSTRAUSS, Extensions of Lipschitz mappings into a Hilbert space, in Conference on Modern Analysis and Probability (1982), pp. 189-206, A. M. S., Providence, RI, 1984.
32. H. KARLOFF, Y. RABANI, and Y. RAVID, Lower bounds for randomized k-server and motion-planning algorithms, SIAM J. Comput. 23 (1994), 293-312.
33. 1, Geometric Funct. Anal. 14 (2004), 745-747.
34. A. J. LEMIN, Isometric embedding of ultrametric (non-Archimedean) spaces in Hilbert space and Lebesgue space, in p-adic Functional Analysis (Loannina, 2000), Lecture Notes in Pure and Appl. Math. 222 (2001), 203-218, Dekker, New York.
35. VLADIMIR I. LEVENSHTEIN, Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, IEEE Trans. Inform. Theory 41 (1995), 1303-1321.
36. N. LINIAL, Finite metric spaces - combinatorics, geometry and algorithms, Proc. of the ICM (Beijing, 2002), 573-586, Higher Ed. Press, Beijing, 2002.
37. N. LINIAL, E. LONDON, and Y. RABINOVICH, The geometry of graphs and some of its algorithmic applications, Combinatorica 15 (1995), 215-245.
38. N. LINIAL, A. MAGEN, and A. NAOR, Girth and euclidean distortion, Geom. Funct. Anal. 12 (2002), 380-394.
39. L. H. LOOMIS and H. WHITNEY, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc. 55 (1949), 961-962.
40. F. J. MACWILLIAMS and N. J. A. SLOANE, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library 16, North-Holland Publ. Co., Amsterdam, 1977.
41. p spaces, Israel J. Math. 102 (1997), 189-197.
42. _, Lectures on Discrete Geometry, Springer-Verlag, New York, 2002.
43. B. MAUREY and G. PISIER, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), 45-90.
44. V. D. MILMAN, A new proof of A. Dvoretzky's theorem on cross-sections of convex bodies, Funk. Anal, i Priložen 5 (1971), 28-37.
45. V. MILMAN and G. SCHECHTMAN, An "isomorphic" version of Dvoretzky's theorem. II, in Convex Geometric Analysis (Berkeley, CA, 1996), pp. 159-164, Cambridge Univ. Press, Cambridge, 1999.
46. _, Asymptotic Theory of Finite-Dimensional Normed Spaces (with an appendix by M. Gromov), Springer-Verlag, New York, 1986.
47. _, An "isomorphic" version of Dvoretzky's theorem, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 541-544.
48. G. PISIER, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge, 1989.
49. F. P. RAMSEY, On a problem of formal logic, Proc. London Math. Soc. 48 (1930), 122-160.
50. S. RAO, Small distortion and volume-preserving embeddings for planar and Euclidean metrics, in Proc. of the Fifteenth Annual Symposium on Computational Geometry (Miami Beach, FL, 1999), pp. 300-306, ACM, New York, 1999 (electronic).
51. N. ROBERTSON and P. D. SEYMOUR, Graph minors. VIII. A Kuratowski theorem for general surfaces, J. Combin. Theory Ser. B 48 (1990), 255-288.
52. J. H. WELLS and L. R. WILLIAMS, Embeddings and Extensions in Analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84, Springer-Verlag, New York, 1975.