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**Zentralblatt MATH**

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

**Zbl.No: ** 097.04202

**Autor: ** Erdös, Pál; Specker, E.

**Title: ** On a theorem in the theory of relations and a solution of a problem of Knaster. (In English)

**Source: ** Colloq. Math. 8, 19-21 (1961).

**Review: ** Let S be a non void set; let f: S ––> PS be a mapping such that for every point x in S fx be a non-void subset of S; if x,y in S and x \not in fy, y \not in fx the points x,y are said independent. A subset X of S is independent in respect to f provided every pair of distinct points of S are independent.

Theorem: If S = I\omega_{m} and if there is some ordinal \beta < \omega_{m} such that for every x in S the set fx is of an ordinal type < \beta, then there exists an independent subset of S of cardinality kS. (The theorem is a generalization of a similar statement in which a uniform boundedness of the cardinals kfx (x in S) intervenes.) The theorem is used to prove a conjecture of Knaster stating that there is no partition A \cup B = R^{2} of plane R^{2} such that every x-line would intersect the set A at a subset of ordinal number < \beta and that every y-line would intersect B at a subset of ordinality < \beta; here one supposes give a normal well-order of the real continuum R inducing a determed well-order of any subset of R.

**Reviewer: ** G.Kurepa

**Classif.: ** * 04A05 Relations, functions

**Index Words: ** set theory

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