ramsey数简介(英文)
来源:互联网 发布:广东网络发票管理系统 编辑:程序博客网 时间:2024/05/18 03:53
Ramsey Number
The Ramsey number
By symmetry, it is true that
(1)
It also must be true that
(2)
A generalized Ramsey number is written
(3)
and is the smallest integer
Bounds are given by
(4)
and
(5)
(Chung and Grinstead 1983). Erdos proved that for diagonal Ramsey numbers
(6)
This result was subsequently improved by a factor of 2 by Spencer (1975).
(7)
Burr (1983) gives Ramsey numbers for all 113 graphs with no more than 6 graph edges and no isolated points.
A summary of known results up to 1983 for
Reference
3
3
6
3
4
9
3
5
14
3
6
18
Graver and Yackel 1968
3
7
23
Kalbfleisch 1966
3
8
28
McKay and Min 1992
3
9
36
Grinstead and Roberts 1982
3
10
[40, 43]
Exoo 1989c, Radziszowski and Kreher 1988
3
11
[46, 51]
Radziszowski and Kreher 1988
3
12
[52, 59]
Exoo 1993, Radziszowski and Kreher 1988, Exoo 1998, Lesser 2001
3
13
[59, 69]
Piwakowski 1996, Radziszowski and Kreher 1988
3
14
[66, 78]
Exoo (unpub.), Radziszowski and Kreher 1988
3
15
[73, 88]
Wang and Wang 1989, Radziszowski (unpub.), Lesser 2001
3
16
[79, 135]
Wang and Wang 1989
3
17
[92, 152]
Wang et al. 1994
3
18
[98, 170]
Wang et al. 1994
3
19
[106, 189]
Wang et al. 1994
3
20
[109, 209]
Wang et al. 1994
3
21
[122, 230]
Wang et al. 1994
3
22
[125, 252]
Wang et al. 1994
3
23
[136, 275]
Wang et al. 1994
4
4
18
4
5
25
McKay and Radziszowski 1995
4
6
[35, 41]
Exoo (unpub.), McKay and Radziszowski 1995
4
7
[49, 61]
Exoo 1989a, Mackey 1994
4
8
[56, 84]
Exoo 1998, Exoo 2002
4
9
[69, 115]
Radziszowski 1988, Mackey 1994
4
10
[92, 149]
Piwakowski 1996, Mackey 1994, Harboth and Krause 2003
4
11
[97, 191]
Piwakowski 1996, Spencer 1994, Burr et al. 1989
4
12
[128, 238]
Su et al. 1998, Spencer 1994
4
13
[133, 291]
Xu and Xie 2002
4
14
[141, 349]
Xu and Xie 2002
4
15
[153, 417]
Xu and Xie 2002
4
16
[153, 815]
4
17
[182, 968]
Luo et al. 2001
4
18
[182, 1139]
4
19
[198, 1329]
Luo et al. 2002
4
20
[230, 1539]
Su et al. 1999
4
21
[242, 1770]
Su et al. 1999
4
22
[282, 2023]
Su et al. 1999
5
5
[43, 49]
Exoo 1989b, McKay and Radziszowski 1995
5
6
[58, 87]
Exoo 1993, Walker 1971
5
7
[80, 143]
CET, Spencer 1994
5
8
[101, 216]
Piwakowski 1996, Spencer 1994, Harborth and Krause 2003
5
9
[121, 316]
Exoo 1998, Haanpää 2000
5
10
[141, 442]
Exoo 1998, Mackey 1994
5
11
[157, 1000]
Exoo 1998, Xiaodong et al. 2004
5
12
[181, 1364]
Exoo 1998
5
13
[205, 1819]
Exoo 1998, Xiaodong et al. 2004
5
14
[233, 2379]
Exoo 1998, Xiaodong et al. 2004
5
15
[261, 3059]
Su et al. 2002, Xiaodong et al. 2004
5
16
[278, 3875]
Luo et al. 2001
5
17
[284, 4844]
Exoo 2002
5
18
[284, 5984]
5
19
[338, 7314]
Su et al. 1999
5
20
[380, 8854]
Luo et al. 2001
5
21
[380, 10625]
5
22
[422, 12649]
Luo et al. 2000
5
23
[434, 14949]
Luo et al. 2000
5
24
[434, 17549]
5
25
[434, 20474]
5
26
[464, 23750]
6
6
[102, 165]
Kalbfleisch 1965, Mackey 1994
6
7
[111, 298]
Exoo 1998, Xu and Xie 2002
6
8
[127, 495]
Exoo 1998, Xu and Xie 2002
6
9
[169, 780]
Exoo 1998, Mackey 1994, Xiaodong et al. 2004
6
10
[178, 1171]
Xu and Xie 2002
6
11
[253, 3002]
Xu and Xie 2002
6
12
[262, 4367]
Xu and Xie 2002
6
13
[317, 6187]
Xu and Xie 2002, Xiaodong et al. 2004
6
14
[317, 8567]
Xu and Xie 2002
6
15
[401, 11627]
Su et al. 2002, Xiaodong et al. 2004
6
16
[434, 15503]
Su et al. 2002
6
17
[548, 20348]
Su et al. 2002
6
18
[614, 26333]
Su et al. 2002
6
19
[710, 33648]
Su et al. 2002
6
20
[878, 42503]
Su et al. 2002
6
21
[878, 53129]
6
22
[1070, 65779]
Su et al. 2002
7
7
[205, 540]
Hill and Irving 1982, Giraud 1973
7
8
[216, 1031]
Xu and Xie 2002
7
9
[232, 1713]
Huang and Zhang 1998, Xiaodong and Zheng 2002
7
10
[232, 2826]
Mackey 1994
7
11
[405, 8007]
Xu and Xie 2002, Xiaodong and Zheng 2002
7
12
[416, 12375]
Xu and Xie 2002
7
13
[511, 18563]
Xu and Xie 2002
7
14
[511, 27131]
7
15
[511, 38759]
7
16
[511, 54263]
7
17
[628, 74612]
Xu and Xie 2002
7
18
[722, 100946]
Xu and Xie 2002
7
19
[908, 134595]
Su et al. 2002
7
20
[908, 177099]
7
21
[1214, 230229]
Su et al. 2002
8
8
[282, 1870]
Burling and Reyner 1972, Mackey 1994
8
9
[317, 3583]
Radziszowski 2002, Xiaodong et al. 2004
8
10
[377, 6090]
Xu and Xie 2002, Huang and Zhang 1998, Xiaodong et al. 2004
8
11
[377, 19447]
8
12
[377, 31823]
8
13
[817, 50387]
Xu and Xie 2002, Xiaodong et al. 2004
8
14
[817, 77519]
8
15
[861, 116279]
Xu and Xie 2002, Xiaodong et al. 2004
8
16
[861, 170543]
8
17
[861, 245156]
Xu and Xie 2002
8
18
[871, 346103]
Xu and Xie 2002
8
19
[1054, 480699]
Xu and Xie 2002
8
20
[1094, 657799]
Su et al. 2002
8
21
[1328, 888029]
Su et al. 2002
9
9
[565, 6588]
Shearer 1986, Shi and Zheng 2001
9
10
[580, 12677]
Xu and Xie 2002
10
10
[798, 23556]
Shearer 1986, Shi 2002
11
11
[1597, 184755]
Mathon 1987
12
12
[1637, 705431]
Xu and Xie 2002
13
13
[2557, 2704155]
Mathon 1987
14
14
[2989, 10400599]
Mathon 1987
15
15
[5485, 40116599]
Mathon 1987
16
16
[5605, 155117519]
Mathon 1987
17
17
[8917, 601080389]
Luo et al. 2002
18
18
[11005, 2333606219]
Luo et al. 2002
19
19
[17885, 9075135299]
Luo et al. 2002
REFERENCES:
Burling, J. P. and Reyner, S. W. "Some Lower Bounds of the Ramsey Numbers
Burr, S. A. "Generalized Ramsey Theory for Graphs--A Survey." In Graphs and Combinatorics (Ed. R. A. Bari and F. Harary).
Burr, S. A. "Diagonal Ramsey Numbers for Small Graphs." J. Graph Th. 7, 57-69, 1983.
Burr, S. A.; Erdos, P.; Faudree, R. J.; and Schelp, R. H. "On the Difference between Consecutive Ramsey Numbers." Util. Math. 35, 115-118, 1989.
Chartrand, G. "The Problem of the Eccentric Hosts: An Introduction to Ramsey Numbers." §5.1 in Introductory Graph Theory.
Chung, F. R. K. "On the Ramsey Numbers
Chung, F. and Grinstead, C. G. "A Survey of Bounds for Classical Ramsey Numbers." J. Graph. Th. 7, 25-37, 1983.
Cipra, B. "A Visit to Asymptopia Yields Insights into Set Structures." Science 267, 964-965, 1995.
Exoo, G. "Applying Optimization Algorithm to Ramsey Problems." In Graph Theory, Combinatorics, Algorithms, and Applications (Ed. Y. Alavi).
Exoo, G. "A Lower Bound for
Exoo, G. "On Two Classical Ramsey Numbers of the Form
Exoo, G. "Announcement: On the Ramsey Numbers
Exoo, G. "A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers of
Exoo, G. "Some New Ramsey Colorings." Electronic J. Combinatorics 5, No. 1, R29, 1-5, 1998.
Exoo, G. "Some Applications of
Folkmann, J. "Notes on the Ramsey Number
Fredricksen, H. "Schur Numbers and the Ramsey Numbers
Gardner, M. "Mathematical Games: In Which Joining Sets of Points by Lines Leads into Diverse (and Diverting) Paths." Sci. Amer. 237, 18-28, 1977.
Giraud, G. "Une minoration du nombre de quadrangles unicolores et son application a la majoration des nombres de Ramsey binaires bicolors." C. R. Acad. Sci. Paris A 276, 1173-1175, 1973.
Graham, R. L.; Rothschild, B. L.; and Spencer, J. H. Ramsey Theory, 2nd ed.
Graver, J. E. and Yackel, J. "Some Graph Theoretic Results Associated with Ramsey's Theorem." J. Combin. Th. 4, 125-175, 1968.
Griggs, J. R. "An Upper Bound on the Ramsey Numbers
Grinstead, C. M. and Roberts, S. M. "On the Ramsey Numbers
Guldan, F. and Tomasta, P. "New Lower Bounds of Some Diagonal Ramsey Numbers." J. Graph. Th. 7, 149-151, 1983.
Haanpää, H. "A Lower Bound for a Ramsey Number." Congr. Numer. 144, 189-191, 2000.
Hanson, D. "Sum-Free Sets and Ramsey Numbers." Discrete Math. 14, 57-61, 1976.
Harary, F. "Recent Results on Generalized Ramsey Theory for Graphs." In Graph Theory and Applications: Proceedings of the Conference at Western Michigan University, Kalamazoo, Mich., May 10-13, 1972 (Ed. Y. Alavi, D. R. Lick, and A. T. White).
Harborth, H. and Krause, S. "Ramsey Numbers for Circulant Colorings." it Congr. Numer. 161, 139-150, 2003.
Hill, R. and Irving, R. W. "On Group Partitions Associated with Lower Bounds for Symmetric Ramsey Numbers." European J. Combin. 3, 35-50, 1982.
Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth.
Huang, Y. R. and Zhang, K. M. "An New Upper Bound Formula for Two Color Classical Ramsey Numbers." J. Combin. Math. Combin. Comput. 28, 347-350, 1998.
Kalbfleisch, J. G. Chromatic Graphs and Ramsey's Theorem. Ph.D. thesis,
Lesser, A. "Theoretical and Computational Aspects of Ramsey Theory." Examensarbeten i Matematik, Matematiska Institutionen, Stockholms Universitet 3, 2001.
Luo, H.; Su, W.; and Li, Z. "The Properties of Self-Complementary Graphs and New Lower Bounds for Diagonal Ramsey Numbers." Australasian J. Combin. 25, 103-116, 2002.
Luo, H.; Su, W.; and Shen, Y.-Q. "New Lower Bounds of Ten Classical Ramsey Numbers." Australasian J. Combin. 24, 81-90, 2001.
Luo, H.; Su, W.; Zhang, Z.; and Li, G. "New Lower Bounds for Twelve Classical 2-Color Ramsey Numbers
Mackey, J. Combinatorial Remedies. Ph.D. thesis. Department of Mathematics,
Mathon, R. "Lower Bounds for Ramsey Numbers and Association Schemes." J. Combin. Th. Ser. B 42, 122-127, 1987.
McKay, B. D. and Min, Z. K. "The Value of the Ramsey Number
McKay, B. D. and Radziszowski, S. P. "
Piwakowski, K. "Applying Tabu Search to Determine New Ramsey Numbers." Electronic J. Combinatorics 3, No. 1, R6, 1-4, 1996.
Radziszowski, S. P. "Small Ramsey Numbers." Electronic J. Combinatorics Dynamical Survey DS1, 1-42, Jul. 4, 2004.
Radziszowski, S. and Kreher, D. L. "Search Algorithm for Ramsey Graphs by
Radziszowski, S. and Kreher, D. L. "Upper Bounds for Some Ramsey Numbers
Robertson, A. "New Lower Bounds for Some Multicolored Ramsey Numbers." Electronic J. Combinatorics 6, No. 1, R3, 1-6, 1999..
Shearer, J. B. "Lower Bounds for Small Diagonal Ramsey Numbers." J. Combin. Th. Ser. A 42, 302-304, 1986.
Shi, L. S. "Upper Bounds for Ramsey Numbers." Preprint. 2002.
Shi, L. S. and Zhang, K. M. "An Upper Bound Formula for Ramsey Numbers" Preprint. 2001.
Spencer, J. H. "Ramsey's Theorem--A New Lower Bound." J. Combinat. Theory Ser. A 18, 108-115, 1975.
Spencer, T. "Upper Bounds for Ramsey Numbers via Linear Programming." Preprint. 1994.
Su, W.; Luo, H.; Li, G.; and Li, Q. "Lower Bounds of Ramsey Numbers Based on Cubic Residues." Disc. Math. 250, 197-209, 2002.
Su, W.; Luo, H.; Li, G.; and Li, Q. "New Lower Bounds of Classical Ramsey Numbers
Su, W.; Luo, H.; Zhang, Z.; and Li, G. "New Lower Bounds of Fifteen Classical Ramsey Numbers." Australasian J. Combin. 19, 91-99, 1999.
Wang, Q. and Wang, G. "New Lower Bounds for the Ramsey Numbers
Wang, Q.; Wang, G.; and Yan, S. "A Search Algorithm and New Lower Bounds for Ramsey Numbers
Whitehead, E. G. "The Ramsey Number
Xiaodong, X. and Zheng, X. "A Constructive Approach for the Lower Bounds on the Ramsey Numbers
Xiaodong, X.; Zheng, X.; Exoo, G.; and Radziszowski, S. P. "Constructive Lower Bounds on Classical Multicolor Ramsey Numbers." Elec. J. Combin. 11, 2004.
- ramsey数简介(英文)
- 圈的ramsey数研究
- XPCOM 简介(英文)
- Instability(Ramsey定理+快速幂)
- hdu6152(拉姆齐(Ramsey)定理) Friend-Graph
- Hdu 6152 Friend-Graph(Ramsey定理)
- HDU 5917 Instability(Ramsey定理)
- ramsey定理
- Ramsey理论
- Ramsey定理
- Ramsey定理
- 英文简短简介一篇
- 2017CCPC网络赛 Friend-Graph(暴力、拉姆齐(Ramsey)定理)
- Small Ramsey Numbers
- 数学中的Ramsey原理
- hdu5917 Instability(ramsey定理)
- HDU 5917:Ramsey定理
- 世界知名英文软件产品简介
- 在Linux下实现comer的TCP/IP协议栈——缓冲池管理和信号灯控制
- 子网掩码的简便算法
- 网站链接有效性判断
- 平面设计常用制作尺寸
- 个人职业生涯规划发展
- ramsey数简介(英文)
- 常用Javascript方法收集
- Ant学习指南
- Javascript之cookie篇
- showModalDialog()的提交弹出新网页的解决方法
- 一些很有用的页面限制代码
- ramsey定理
- 打开,另存为,属性,打印"等14个JS代码
- javascript 常用类