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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 218.52005

**Autor: ** Erdös, Paul; Méir, A.; Sós, V.T.; Turán, P.

**Title: ** On some applications of graph theory. II. (In English)

**Source: ** Stud. Pure Math., Papers presented to Richard Rado on the Occasion of his sixtyfifth Birthday, 89-99 (1971).

**Review: ** [For the entire collection see Zbl 214.00204.]

Using combinatorial methods the authors prove (among others) the following theorem: Let there be given n points in the plane P_{1}, ... ,P_{n} so that the maximal area of all triangles (P_{i},P_{j},P_{l}) is 1. Then at least ^{1}/_{7} \binom{n}{3} of these triangles have an area \leq {\sqrt 5-1 \over 2}. The authors conjectured and B. Bollobás proved that if n = 4m there is an absolute constant c so that at most 4m^{3} of the triangles can have area > 1-c. 4m^{3} is best possible, but the best value of c is not known. They also show that if n = 5 at least one of the triangles have area \leq {\sqrt 5 -1 \over 2}. (The regular pentagon P_{1},P_{2},P_{3},P_{4},P_{5}, area (P_{1},P_{3},P_{4}) is 1 shows that {\sqrt 5 -1 \over 2} is best possible). Many unsolved problems remain.

**Classif.: ** * 52B05 Combinatorial properties of convex sets

05C99 Graph theory

**Citations: ** Zbl 214.00204(EA)

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag