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Turán-Ramsey Theorems and Kp-Independence Numbers

Published online by Cambridge University Press:  12 September 2008

P. Erdős ,
A. Hajnal ,
M. Simonovits ,
V. T. Sós  and
E. Szemerédi
P. Erdős
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest.
A. Hajnal
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest.
M. Simonovits
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest.
V. T. Sós
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest.
E. Szemerédi
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest.
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Abstract

Let the Kp-independence number αp (G) of a graph G be the maximum order of an induced subgraph in G that contains no Kp. (So K2-independence number is just the maximum size of an independent set.) For given integers r, p, m > 0 and graphs L1,…,Lr, we define the corresponding Turán-Ramsey function RTp(n, L1,…,Lr, m) to be the maximum number of edges in a graph Gn of order n such that αp(Gn) ≤ m and there is an edge-colouring of G with r colours such that the jth colour class contains no copy of Lj, for j = 1,…, r. In this continuation of [11] and [12], we will investigate the problem where, instead of α(Gn) = o(n), we assume (for some fixed p > 2) the stronger condition that αp(Gn) = o(n). The first part of the paper contains multicoloured Turán-Ramsey theorems for graphs Gn of order n with small Kp-independence number αp(Gn). Some structure theorems are given for the case αp(Gn) = o(n), showing that there are graphs with fairly simple structure that are within o(n2) of the extremal size; the structure is described in terms of the edge densities between certain sets of vertices.

The second part of the paper is devoted to the case r = 1, i.e., to the problem of determining the asymptotic value of

for p < q. Several results are proved, and some other problems and conjectures are stated.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 3 , September 1994 , pp. 297 - 325
Copyright
Copyright © Cambridge University Press 1994

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References

[1]Bollobás, B. (1989) An extension of the isoperimetric inequality on the sphere. Elemente der Math. 44 121–124.Google Scholar
[2]Bollobás, B. (1978) Extremal graph theory, Academic Press, London.Google Scholar
[3]Bollobás, B. and Erdős, P. (1976) On a Ramsey-Turán type problem. Journal of Combinatorial Theory B 21 166–168.Google Scholar
[4]Brown, W. G., Erdős, P. and Simonovits, M. (1973) Extremal problems for directed graphs. Journal of Combinatorial Theory B 15 (1) 77–93.CrossRefGoogle Scholar
[5]Brown, W. G., Erdős, P. and Simonovits, M. (1978) On multigraph extremal problems. In: J., Bermond et al. (ed.) Problèmes Combinatoires et Thèorie des Graphes, (Proc. Conf. Orsay 1976), CNRS Paris63–66.Google Scholar
[6]Brown, W. G., Erdős, P. and Simonovits, M. (1985) Inverse extremal digraph problems. Finite and Infinite Sets, Eger (Hungary) 1981. Colloq. Math. Soc. J. Bolyai 37, Akad. Kiadó, Budapest119–156.Google Scholar
[7]Brown, W. G., Erdős, P. and Simonovits, M. (1985) Algorithmic Solution of Extremal Digraph Problems. Transactions of the American Math Soc. 292/2421–449.CrossRefGoogle Scholar
[8]Erdős, P. (1968) On some new inequalities concerning extremal properties of graphs. In: Erdős, P. and Katona, G. (ed. ) Theory of Graphs (Proc. Coll. Tihany, Hungary, 1966), Acad. Press N.Y.77–81.Google Scholar
[9]Erdős, P. (1961) Graph Theory and Probability, II. Canad. Journal of Math. 13 346–352.CrossRefGoogle Scholar
[10]Erdős, P. and Hajnal, A. (1966) On chromatic number of graphs and set-systems. Ada Math. Acad. Sci. Hung. 17 61–99.CrossRefGoogle Scholar
[11]Erdős, P., Hajnal, A., Sós, V. T. and Szemerédi, E. (1983) More results on Ramsey-Turán type problems. Combinatorica 3 (1) 69–82.CrossRefGoogle Scholar
[12]Erdős, P., Hajnal, A., Simonovits, M, Sós, V. T. and Szemerédi, E. (1993) Turán-Ramsey theorems and simple asymptotically extremal structures. Combinatorica 13 31–56.CrossRefGoogle Scholar
[13]Erdős, P., Meir, A., Sós, V. T. and Turán, P. (1972) On some applications of graph theory I. Discrete Math. 2 (3) 207–228.CrossRefGoogle Scholar
[14]Erdős, P., Meir, A., Só's, V. T. and Turán, P. (1971) On some applications of graph theory II. Studies in Pure Mathematics (presented to R. Rado), Academic Press, London89–99.Google Scholar
[15]Erdős, P., Meir, A., Sós, V. T. and Turán, P. (1972) On some applications of graph theory III. Canadian Math. Bulletin 15 27–32.Google Scholar
[16]Erdős, P. and Rogers, C. A. (1962) The construction of certain graphs. Canadian Journal of Math 702–707. (Reprinted in Art of Counting, MIT PRESS.)Google Scholar
[17]Erdős, P. and Simonovits, M. (1966) A limit theorem in graph theory. Studia Sci. Math. Hungar. 1 51–57.Google Scholar
[18]Erdős, P. and Só, V. T. (1969) Some remarks on Ramsey's and Turin's theorems. In: Erdős, P. et al. (eds.) Combin. Theory and Appl. Mathem. Coll. Soc. J. Bolyai 4, Balatonfüred395–404.Google Scholar
[19]Erdős, P. and Stone, A. H. (1946) On the structure of linear graphs. Bull. Amer. Math. Soc. 52 1089–1091.CrossRefGoogle Scholar
[20]Frankl, P. and Rödl, V. (1988) Some Ramsey-Turán type results for hypergraphs. Combinatorica 8 (4) 323–332.CrossRefGoogle Scholar
[21]Graham, R. L., Rothschild, B. L. and Spencer, J. (1980) Ramsey Theory, Wiley Interscience, Ser. in Discrete Math.Google Scholar
[22]Katona, G. (1985) Probabilistic inequalities from extremal graph results (a survey). Annals of Discrete Math. 28 159–170.Google Scholar
[23]Ramsey, F. P. (1930) On a problem of formal logic. Proc. London Math. Soc, 2nd Series 30 264–286.CrossRefGoogle Scholar
[24]Shamir, E. (1988) Generalized stability and chromatic numbers of random graphs (preprint, under publication).Google Scholar
[25]Sidorenko, A. F. (1980) Klasszi gipergrafov i verojatnosztynije nyeravensztva, Dokladi 254/3,Google Scholar
[26]Sidorenko, A. F. (1983) (Translation) Extremal estimates of probability measures and their combinatorial nature. Math. USSR - Izv 20 N3 503–533 MR 84d: 60031. (Original: Izvest. Acad. Nauk SSSR. ser. matem. 46 N3 535–568.)CrossRefGoogle Scholar
[27]Sidorenko, A. F. (1989) Asymptotic solution for a new class of forbidden r-graphs. Combinatorica 9 (2) 207–215.CrossRefGoogle Scholar
[28]Simonovits, M. (1968) A method for solving extremal problems in graph theory. In: Erdős, P. and Katona, G. (ed.) Theory of Graphs (Proc. Coll. Tihany, Hungary, 1966), Acad. Press N. Y. 279–319.Google Scholar
[29]Simonovits, M. (1983) Extremal Graph Theory. In: Beineke, and Wilson, (ed.) Selected Topics in Graph Theory, Academic Press, London, New York, San Francisco161–200.Google Scholar
[30]Sós, V. T. (1969) On extremal problems in graph theory. Proc. Calgary International Conf. on Combinatorial Structures and their Application407–410.Google Scholar
[31]Szemerédi, E. (1972) On graphs containing no complete subgraphs with 4 vertices (in Hungarian). Mat. Lapok 23 111–116.Google Scholar
[32]Szemerédi, E. (1978) On regular partitions of graphs. In: Bermond, J. et al. . (ed.) Problèmes Combinatoires et Théorie des Graphes, (Proc. Conf. Orsay 1976), CNRS Paris 399–401.Google Scholar
[33]Turán, P. (1941) On an extremal problem in graph theory (in Hungarian). Matematikai Lapok 48 436–452.Google Scholar
[34]Turán, P. (1954) On the theory of graphs. Colloq. Math. 3 19–30.CrossRefGoogle Scholar
[35]Turán, P. (1969) Applications of graph theory to geometry and potential theory. In: Proc. Calgary International Conf. on Combinatorial Structures and their Application423–434.Google Scholar
[36]Turán, P. (1972) Constructive theory of functions. Proc. Internat. Conference in Varna, Bulgaria, 1970, Izdat. Bolgar Akad. Nauk, Sofia.Google Scholar
[37]Turán, P. (1970) A general inequality of potential theory. Proc. Naval Research Laboratory, Washington137–141.Google Scholar
[38]Turán, P. (1989) Collected papers of Paul Turán Vol 1–3, Akadé;miai Kiadò, Budapest.Google Scholar
[39]Zykov, A. A. (1949) On some properties of linear complexes. Mat Sbornik 24 163–188. (Amer. Math. Soc. Translations 79 (1952)).Google Scholar