## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  499.05014
Autor:  Erdös, Paul; Larson, J.
Title:  On pairwise balanced block designs with the sizes of blocks as uniform as possible. (In English)
Source:  Ann. Discrete Math. 15, 129-134 (1982).
Review:  A pairwise balanced design on a finite set S of n elements is a collection, L, of subsets of S with the property that every 2-subset of S is contained in a unique member of L. If every member of L is of the same cardinality, m+1 say, then we have either a trivial situation, m = 1 or m = n-1, or we have a finite projection plane of order m. The authors seek to find pairwise balanced designs with |L| approximately \sqrt{n} for every L in \Cal{L}. They show, both constructively and probabilistically, how to insure that |L| = \sqrt{n}+O(n ½ -c) for every L in \Cal{L}, where c is a fixed constant, n arbitrary. Their proof utilizes the notion of an arc in a finite projective plane. The authors appear to be unaware of the work on arcs done by the Italian school and others.
Reviewer:  E.F.Assmus jun
Classif.:  * 05B05 Block designs (combinatorics)
05B25 Finite geometries (combinatorics)
Keywords:  arcs in projective planes; pairwise balanced designs; finite projective plane

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