Ramsey Number

The Ramsey number  gives the solution to the party problem, which asks the minimum number of guests  that must be invited so that at least  will know each other or at least  will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices  such that all undirected simple graphs of order  contain a clique of order  or an independent set of order . Ramsey's theorem states that such a number exists for all  and .

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  such that, no matter how each -element subset of an -element set is colored with  colors, there exists an  such that there is a subset of size , all of whose -element subsets are color . The usual Ramsey numbers are then equivalent to .

Bounds are given by

(4)

and

(5)

(Chung and Grinstead 1983). Erdős proved that for diagonal Ramsey numbers ,

(6)

This result was subsequently improved by a factor of 2 by Spencer (1975).  was known since 1980 to be bounded from above by , and Griggs (1983) showed that  was an acceptable limit. J.-H. Kim (Cipra 1995) subsequently bounded  by a similar expression from below, so

(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  is given in Chung and Grinstead (1983). Radziszowski (2004) maintains an up-to-date list of the best current bounds. Results from Tables I and II of Radziszowski (2004) are reproduced below in a slightly less cramped format than in the original. Known bounds for generalized Ramsey numbers (multicolor graph numbers), hypergraph Ramsey numbers, and many other types of Ramsey numbers may be found in Radziszowski (2004). In the absence of a published upper bound, the theorem of Erdős-Szekeres stating that  is used to provide one.

			Reference
3	3	6	Greenwood and Gleason 1955
3	4	9	Greenwood and Gleason 1955
3	5	14	Greenwood and Gleason 1955
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	Greenwood and Gleason 1955
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	[73, 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	[125, 316]	Exoo 1998, Haanpää 2000
5	10	[143, 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	[113, 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	[179, 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	[233, 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 ." J. Combin. Th. Ser. B 13, 168-169, 1972.

Burr, S. A. "Generalized Ramsey Theory for Graphs--A Survey." In Graphs and Combinatorics (Ed. R. A. Bari and F. Harary). New York: Springer-Verlag, pp. 52-75, 1974.

Burr, S. A. "Diagonal Ramsey Numbers for Small Graphs." J. Graph Th. 7, 57-69, 1983.

Burr, S. A.; Erdős, 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. New York: Dover, pp. 108-115, 1985.

Chung, F. R. K. "On the Ramsey Numbers ." Discrete Math. 5, 317-321, 1973.

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). Philadelphia: SIAM, pp. 175-179, 1989a.Exoo, G. "A Lower Bound for ." J. Graph Th. 13, 97-98, 1989.

Exoo, G. "On Two Classical Ramsey Numbers of the Form ." SIAM J. Discrete Math. 2, 488-490, 1989c.Exoo, G. "Announcement: On the Ramsey Numbers ,  and ." Ars Combin. 35, 85, 1993.

Exoo, G. "A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers of ." Electronic J. Combinatorics 1, No. 1, R8, 1-3, 1994.

 http://www.combinatorics.org/Volume_1/Abstracts/v1i1r8.html.Exoo, G. "Some New Ramsey Colorings." Electronic J. Combinatorics 5, No. 1, R29, 1-5, 1998.

 http://www.combinatorics.org/Volume_5/Abstracts/v5i1r29.html.Exoo, G. "Some Applications of -Groups in Graph Theory." Preprint. 2002.

Folkmann, J. "Notes on the Ramsey Number ." J. Combinat. Theory. Ser. A 16, 371-379, 1974.

Fredricksen, H. "Schur Numbers and the Ramsey Numbers ." J. Combin. Theory Ser. A 27, 376-377, 1979.

Gardner, M. "Mathematical Games: In Which Joining Sets of Points by Lines Leads into Diverse (and Diverting) Paths." Sci. Amer. 237, 18-28, 1977.

Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 240-241, 1989.

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. New York: Wiley, 1990.

Graver, J. E. and Yackel, J. "Some Graph Theoretic Results Associated with Ramsey's Theorem." J. Combin. Th. 4, 125-175, 1968.

Greenwood, R. E. and Gleason, A. M. "Combinatorial Relations and Chromatic Graphs." Canad. J. Math. 7, 1-7, 1955.

Griggs, J. R. "An Upper Bound on the Ramsey Numbers ." J. Comb. Th. A 35, 145-153, 1983.

Grinstead, C. M. and Roberts, S. M. "On the Ramsey Numbers  and ." J. Combinat. Th. Ser. B 33, 27-51, 1982.

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). New York: Springer-Verlag, pp. 125-138, 1972.

Harborth, H. and Krause, S. "Ramsey Numbers for Circulant Colorings." 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 Erdős and the Search for Mathematical Truth. New York: Hyperion, pp. 52-53, 1998.

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, University of Waterloo, January 1966.

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 ." Guangxi Sci. 7, 120-121, 2000.

Mackey, J. Combinatorial Remedies. Ph.D. thesis. Department of Mathematics, University of Hawaii, 1994.

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 ." J. Graph Th. 16, 99-105, 1992.

McKay, B. D. and Radziszowski, S. P. "." J. Graph. Th 19, 309-322, 1995.

Piwakowski, K. "Applying Tabu Search to Determine New Ramsey Numbers." Electronic J. Combinatorics 3, No. 1, R6, 1-4, 1996.

 http://www.combinatorics.org/Volume_3/Abstracts/v3i1r6.html.Radziszowski, S. P. "Small Ramsey Numbers." Electronic J. Combinatorics Dynamical Survey DS1, 1-42, Jul. 4, 2004.

 http://www.combinatorics.org/Surveys/#DS1.Radziszowski, S. and Kreher, D. L. "Search Algorithm for Ramsey Graphs by Union of Group Orbits." J. Graph Th. 12, 59-72, 1988a.Radziszowski, S. and Kreher, D. L. "Upper Bounds for Some Ramsey Numbers ." J. Combinat. Math. Combin. Comput. 4, 207-212, 1988b.Robertson, A. "New Lower Bounds for Some Multicolored Ramsey Numbers." Electronic J. Combinatorics 6, No. 1, R3, 1-6, 1999.

 http://www.combinatorics.org/Volume_6/Abstracts/v6i1r3.html.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 , , and ." Chinese Sci. Bull. 43, 528, 1998.

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 ." Beijing Daxue Xuebao 25, 117-121, 1989.

Wang, Q.; Wang, G.; and Yan, S. "A Search Algorithm and New Lower Bounds for Ramsey Numbers ." Preprint. 1994.Whitehead, E. G. "The Ramsey Number ." Discrete Math. 4, 389-396, 1973.

Xiaodong, X. and Zheng, X. "A Constructive Approach for the Lower Bounds on the Ramsey Numbers ." Unpublished manuscript, 2002.Xiaodong, X.; Zheng, X.; Exoo, G.; and Radziszowski, S. P. "Constructive Lower Bounds on Classical Multicolor Ramsey Numbers." Elec. J. Combin. 11, 2004. http://www.combinatorics.org/Volume_11/PDF/v11i1r35.pdf.