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Paul Erdős’ Influence on Extremal Graph Theory

  • Chapter
The Mathematics of Paul Erdös II

Part of the book series: Algorithms and Combinatorics ((AC,volume 14))

Summary

Paul Erdős is 801 and the mathematical community is celebrating him in various ways. Jarik Neřetšil also organized a small conference in Prague in his honour, where we, combinatorists and number theorists attempted to describe in a limited time the enourmous influence Paul Erdős made on the mathematics of our surrounding (including our mathematics as well). Based on my lecture given there, I shall to survey those parts of Extremal Graph Theory that are connected most directly with Paul Erdős’s work.

In Turán type extremal problems we usually have some sample graphs L 1, …, L r, and consider a graph G n on n vertices not containing any L i . We ask for the maximum number of edges such a G n can have. We may ask similar questions for hypergraphs, multigraphs and digraphs.

We may also ask, how many copies of forbidden subgraphs L i must a graph G n contain with a given number of edges superseding the maximum in the corresponding extremal graph problems. These are the problems on Supersaturated Graphs.

We can mix these questions with Ramsey type problems, (Ramsey-Turán Theory). This topic is the subject of a survey by V. T. Sós [162].

These topics are definitely among the favourite areas in Paul Erdős’s graph theory.

Supported by GRANT “OTKA 1909”.

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  1. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary

    Miklós Simonovits

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  1. AT&T Bell Laboratories, 600 Mountain Avenue, 07974, Murray Hill, NJ, USA

    Ronald L. Graham

  2. Department of Applied Mathematics, Charles University, Malostranske nam. 25, 11800, Praha, Czech Republic

    Jaroslav Nešetřil

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Simonovits, M. (1997). Paul Erdős’ Influence on Extremal Graph Theory. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös II. Algorithms and Combinatorics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60406-5_15

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