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 P1, ... ,Pn so that the maximal area of all triangles (Pi,Pj,Pl) 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 4m3 of the triangles can have area > 1-c. 4m3 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 P1,P2,P3,P4,P5, area (P1,P3,P4) 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)
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