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Generalized ramsey theory for graphs - a survey

  • Part II: Contributed Papers Expository Review Articles
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Graphs and Combinatorics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 406))

Abstract

Almost nonexistent a few years ago, the field of generalized Ramsey theory for graphs is now being pursued very actively and with remarkable success. This survey paper will emphasize the following class of problems: Given graphs G1, ..., Gc, determine or estimate the Ramsey number r(G1, ..., Gc), the smallest number p such that if the lines of a complete graph Kp are c-colored in any manner, then for some j there exists a subgraph in color j which is isomorphic to Gj. Ramsey numbers have now been evaluated completely in a large number of cases, particularly when c = 2. The most strikingly general result is due to Chvátal: If T is a tree on m points, then r(T,Kn) = mn-m-n+2. Also of interest is the study of asymptotic questions about r(G1, ..., Gc). For instance, Burr, Erdös, and Spencer have shown that if G has m points, none of them isolated, and if i is the maximal number of independent points in G, then (2m-i)n-1 ≤ r (nG,nG) ≤ (2m-i)n+C, where C is a constant depending only on G.

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Authors and Affiliations

  1. Bell Laboratories, Madison, N.J.

    Stefan A. Burr

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  1. Stefan A. Burr
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Ruth A. Bari Frank Harary

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© 1974 Springer-Verlag Berlin

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Burr, S.A. (1974). Generalized ramsey theory for graphs - a survey. In: Bari, R.A., Harary, F. (eds) Graphs and Combinatorics. Lecture Notes in Mathematics, vol 406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066435

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  • DOI: https://doi.org/10.1007/BFb0066435

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  • Print ISBN: 978-3-540-06854-9

  • Online ISBN: 978-3-540-37809-9

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