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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 521.05004

**Autor: ** Erdös, Paul; Silverman, R.; Stein, A.

**Title: ** Intersection properties of families containing sets of nearly the same size. (In English)

**Source: ** Ars Comb. 15, 247-259 (1983).

**Review: ** This paper addresses the question: one wants to choose a set S of points in a projective plane of order n such that every line contains at least one point of S, and such that no line has more than x members of S. How small can x be: It is shown that x can be chosen of the order c log n. One can argue as follows: if one chooses every point with probability P at random, the distribution of the number of points on any line obeys a binomial distribution and can be explicitly evaluated. If p is chosen such that the probability of obtaining either 0 or x or more points on a line is less than **(**\frac{1}{n^{2}+n+1}**)** or **(**\frac{1}{n^{2}+n+1}**)**^{-1}, then there is a finite probability that no line in the plane falls to intersect the chosen points in at least one, but x or fewer, points. This paper contains detailed computations showing that this occurs for x = c log n and appropriate c. It also contains discussion of a less efficient constructive example.

**Reviewer: ** D.Kleitman

**Classif.: ** * 05A05 Combinatorial choice problems

05B25 Finite geometries (combinatorics)

60C05 Combinatorial probability

**Keywords: ** families of finite sets; projective plane; binomial distribution

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