Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 732.51007
Autor: Drake, David A.; Erdös, Paul
Title: Bounds on the number of pairs of unjoined points in a partial plane. (In English)
Source: Coding theory and design theory. Part II: Design theory, Proc. Workshop IMA Program Appl. Comb., Minneapolis/MN (USA) 1987-88, IMA Vol. Math. Appl. 21, 102-112 (1990).
Review: [For the entire collection see Zbl 693.00019.]
A partial plane is a pair \Sigma = (S,C) where S is a finite set (of points) and C is a collection of subsets of S (whose elements are called lines) such that no two points of \Sigma lie on more than one common line. The deficit D of a partial plane is the number of pairs of unjoined points in \Sigma. With v = |S|, b = |C| and k denoting the cardinality of the largest line in \Sigma, the principal result is:
Theorem: Let n be an integer such that n \geq 29, n2-n+3 \leq v \leq n2+n-1, b < v, k \leq v-3. Then 2D > 3v-13v3/4.
This and other bounds given in the paper may be loosely phrased as saying that if a partial plane \Sigma has a small deficit then b is large relative to v. The proofs proceed by considering separately the cases when k is small or large relative to v. In the large line case linear programming techniques are utilized.
Reviewer: W.E.Cherowitzo (Denver)
Classif.: * 51E14 Finite partial geometries (general), nets, fractial spreads
05B30 Other designs, configurations
90C05 Linear programming
Keywords: linear space; projective plane; partial plane
Citations: Zbl 693.00019
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