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**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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