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Maximal coplanar sets of intersection points

Published online by Cambridge University Press:  17 April 2009

Richard P. Halpern ,
David Hobby  and
Donald M. Silberger
Richard P. Halpern
Affiliation:
State University of New York, New Paltz, NY 12561, United States of America
David Hobby
Affiliation:
State University of New York, New Paltz, NY 12561, United States of America
Donald M. Silberger
Affiliation:
State University of New York, New Paltz, NY 12561, United States of America Universidade Federal de Santa Catarina, 88.000 - Florianópolis – SC, Brasil
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Abstract

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Let F be any set of five points in R3 so situated that no four of the points are coplanar, and that the line xy through any two x and y of the points has a unique intersection point xy* with the plane determined by the other three. Let F^ denote the family of all such xy*. Let S(F) denote the set of all X ⊆ F^ which are maximal with respect to the property that X is a subset of a plane in R3. For k > 2 an integer, let S(k; F) denote the family of all k-membered elements in S(F).

A family 𝒟 of sets is said to be uniformly deep of depth d if and only if for every x ∈ ∪ 𝒟 there are exactly d distinct 𝒜 ∈ 𝒟 for which x ∈ 𝒜.

We establish the following result, and extend our ideas to general Euclidean spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 42 , Issue 1 , August 1990 , pp. 41 - 56
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Harary, Frank, Graph Theory (Addison-Wesley, Reading, Mass., 1972).Google Scholar
[2]Silberger, D.M., ‘Uniformly deep families of k–membered subsets of n’, J. Combin. Theory Ser. A 22 (1977), 31–37.CrossRefGoogle Scholar