Abstract
This is a progress report on a very dynamic branch of graph theory. We begin with a historical review of the origins of generalized ramsey theory and then indicate the small graphs for which the diagonal ramsey numbers are now known. The ramsey multiplicity of a graph is taken up and applied to ramsey games. We conclude with a listing of those families of graphs for which the ramsey numbers have been determined. There still does not exist any general powerful method for computing ramsey numbers.
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References
Burr, S. A., Generalized Ramsey Theory for Graphs — a Survey, this volume p. 52
Burr, S. A., Diagonal Ramsey Numbers for Small Graphs, to appear.
Burr, S. A., and Roberts, J. A., On Ramsey Numbers for Linear Forests, Discrete Math, to appear.
Chvátal, V., Tree — Complete Graph Ramsey Numbers, J. Combinatorial Theory, to appear.
Chvátal, V., and Clancy, M., Diagonal Ramsey Numbers for Most 5-point Graphs, to appear.
Chvátal, V., and Harary, F., Generalized Ramsey Theory for Graphs, Bull. Amer. Math. Soc. 78 (1972) 423–426.
____, Generalized Ramsey Theory for Graphs I, Diagonal Numbers, Periodica Math. Hungar. 3(1973) 113–122.
____, Generalized Ramsey Theory for Graphs, II, Small Diagonal Numbers, Proc. Amer. Math. Soc. 32 (1972) 389–394.
____, Generalized Ramsey Theory for Graphs, III, Small Off-Diagonal Numbers, Pacific J. Math. 41 (1972) 335–345.
Cockayne, E. J., An Application of Ramsey's Theorem, Canad. Math. Bull. 13 (1970) 145–146.
Cockayne, E. J., and Lorimer, P. J., On Ramsey Numbers for Stars and Stripes, Canad. Math. Bull., to appear.
Erdös, P., Applications of Probabilistic Methods to Graph Theory, A Seminar on Graph Theory (F. Harary, ed.) Holt, 1967, 60–64.
Faudree, R. J., Lawrence, S. L., Parsons, T. D., and Schelp, R. H., Path-cycle Ramsey Numbers, to appear.
Faudree, R. J., and Schelp, R. H., All Ramsey Numbers for Cycles in Graphs, to appear.
Gardner, M., Mathematical Games, Scientific American 228 (January, 1973) 108–111.
Gerencśer, L., and Gyárfás, A., On Ramsey-type Problems, Ann. Univ. Sci. Budapest Eötvös 10 (1967) 167–170.
Goodman, A. W., On Sets of Acquaintances and Strangers at any Party, Amer. Math. Monthly 66 (1959) 778–783.
Harary, F., Graph Theory, Addison-Wesley, Reading, 1969.
Harary, F., The Two-Triangle Case of the Acquaintance Graph, Math. Mag. 45 (1972) 130–135.
Harary, F., Recent Results on Generalized Ramsey Theory for Graphs, Graph Theory and Applications (Y. Alavi, et al. eds.), Springer, Berlin, 1972, 125–138.
Harary, F., and Hell, P., Generalized Ramsey Theory for Graphs, V, Ramsey Numbers for Digraphs, Bull. London Math. Soc., to appear.
Harary, F., and Prins, G., Generalized Ramsey Theory for Graphs IV, Ramsey Multiplicities, Networks, to appear.
Parsons, T. D., Path-star Ramsey Numbers, Proc. Amer. Math. Soc., to appear.
Ramsey, F. P., On a Problem of Formal Logic, Proc. London Math. Soc. 30 (1930) 264–286.
Rosta, V., On a Ramsey Type Problem of J. A. Bondy and P. Erdös, I, II, J. Combinatorial Theory, 15B (1973), 94–104, 105–120.
Schwenk, A. J., Acquaintance Graph Party Problem, Amer. Math. Monthly, 79 (1972) 1113–1117.
Simmons, G. J., The Game of Sim, J. Recreational Math. 2 (1969) 66.
Williamson, J., A Ramsey-type Problem for Paths in Digraphs, Math. Ann. 203 (1973) 117–118.
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© 1974 Springer-Verlag Berlin
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Harary, F. (1974). A survey of generalized ramsey theory. In: Bari, R.A., Harary, F. (eds) Graphs and Combinatorics. Lecture Notes in Mathematics, vol 406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066430
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DOI: https://doi.org/10.1007/BFb0066430
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