Abstract
Virtually all of the known results on generalized Ramsey theory for graphs have been reported here, and the most general method of proof was brute force. There is certainly a need for more powerful and general methods, but it is not certain that these exist. Since the study of Ramsey properties of general graphs appears less intractable than that for complete graphs, this may well suggest fruitful directions for other mathematical structures such as vector spaces. The fact that generalized Ramsey theory for graphs is in its infancy is attested by more than half of the references having the status, “to appear”.
Research supported in part by Grant 68-1515 from the Air Force Office of Scientific Research. The author thanks A. J. Schwenk for his assistance in the preparation of this article.
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References
J. A. Bondy and P. Erdős, Ramsey numbers for cycles in graphs, to appear.
S. A. Burr and J. A. Roberts, On Ramsey numbers for paths, to appear.
__________, On Ramsey numbers for stars, to appear.
G. Chartrand and S. Schuster, On the existence of specified cycles in complementary graphs, Bull. Amer. Math. Soc. 77 (1971), 995–998.
__________, A note on cycle Ramsey numbers, Discrete Math., submitted for publication.
V. Chvátal and F. Harary, Generalized Ramsey theory for graphs, Bull. Amer. Math. Soc., to appear.
V. Chvátal and F. Harary, Generalized Ramsey theory for graphs, I, diagonal numbers, Periodica Math. Hungar., to appear.
__________, 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., to appear.
J. Folkman, Graphs with nonochromatic complete subgraphs in every edge coloring, Studies in Combinatorics (J. B. Kruskel and E. C. Posner, eds.), SIAM, Philadelphia, 1970.
A. W. Goodman, On sets of acquaintances and strangers at any party, Amer. Math. Monthly 66 (1959), 778–783.
R. L. Graham, On edgewise 2-colored graphs with monochromatic triangles and containing no complete hexagon, J. Combinatorial Theory 4 (1968), 300.
R. L. Graham and J. Spencer, On small graphs with forced monochromatic triangles, Recent Trends in Graph Theory, (M. Capobianco et al., eds.), Springer, New York, 1971, 137–141.
F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969.
F. Harary, The two triangle case of the acquaintance graph, Math. Maq. 45 (1972), 130–135.
F. Harary and P. Hell, Generalized Ramsey theory for graphs, IV, Ramsey numbers for digraphs, to appear.
F. Harary and G. Prins, Generalized Ramsey theory for graphs, V, Ramsey multiplicities, to appear.
F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264–286.
S. Schuster, Progress on the problem of eccentric hosts, this volume, 283–290.
A. Schwenk, The acquaintance graph revisited, Amer. Math. Monthly, to appear.
J. Williamson, A Ramsey-type problem for paths in digraphs, to appear.
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Harary, F. (1972). Recent results on generalized Ramsey theory for graphs. In: Alavi, Y., Lick, D.R., White, A.T. (eds) Graph Theory and Applications. Lecture Notes in Mathematics, vol 303. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067364
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DOI: https://doi.org/10.1007/BFb0067364
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