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On Some Applications of Graph Theory III

Published online by Cambridge University Press:  20 November 2018

P. Erdös ,
A. Meir ,
V. T. Sós  and
P. Turán
P. Erdös
Affiliation:
Hungarian Academy of Sciences, Budapest, Hungary
A. Meir
Affiliation:
Hungarian Academy of Sciences, Budapest, Hungary
V. T. Sós
Affiliation:
Hungarian Academy of Sciences, Budapest, Hungary
P. Turán
Affiliation:
University of Alberta, Edmonton, Alberta
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In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of "triangle functional " (e.g. perimeter, area of triangles) [2]. The basic tool is provided in all these applications by the result formulated as Lemma 2. This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications.

Here the same method is used in an abstract setting. First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functional on Hilbert spaces were given by Katona [3].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 1 , March 1972 , pp. 27 - 32
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Erdös, P., Meir, A., Sös, V. T. and Turán, P., On some applications of graph theory I, Discrete Mathematics (to appear).Google Scholar
2. Erdös, P., Meir, A., Sös, V. T. and Turán, P., On some applications of graph theory II, Studies in Pure Mathematics, Academic Press, (1971), 89-100.Google Scholar
3. Katona, Gy., Gráfok, vektorok es valöszinuségszámitási egyenlötlenségek (in Hungarian, with English and Russian abstracts). Mat. Lapok. Vol. 20, Fasc. 1-2 (1969), 123-127.Google Scholar
4. Kolmogorov, A. N. and Tihomirov, V. M., ɛ-entropy and ɛ-capacity of sets in function spaces, Uspehi Mat. Nauk. no. 2 (86), 14 (1959), 3-86; English transi., Amer. Math. Soc. Transi. (2) 17 (1961), 277-364.Google Scholar
5. Newman, D. J. and Raymon, L., Optimally separated contractions, Amer. Math. Monthly 77 (1970), 58-59.Google Scholar
6. Turán, P., Egy gráfelméleti szélsöértékfeladatröl. (Hungarian with German abstract.) Mat. Lapok. 49 (1941), 436-452. Reproduced in English in the Appendix to P. Turán, On the theory of graphs, Colloq. Math., 1954.Google Scholar