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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000­000
S 0002-9939(XX)0000-0
STABLE KNESER HYPERGRAPHS AND
IDEALS IN N WITH THE NIKOD´YM PROPERTY
NOGA ALON, LECH DREWNOWSKI, AND TOMASZ LUCZAK
Abstract. We use stable Kneser hypergraphs to construct ideals in N which
are not nonatomic yet have the Nikod´ym property.
1. Introduction
We say that an ideal I in P(N) has the Positive Summability Property (PSP)
if for any sequence (xn) of positive reals such that

n=1 xn diverges, there exists
I I for which nI xn diverges as well. It is not hard to show that the ideal Z
of sets of density zero has (PSP). This `folklore' fact can be traced back (at least)
to a short note of Auerbach [2] from 1930, and has later been rediscovered several
times (for more information see [3]). However, all known proofs of this and similar
results are based on the fact that the ideal in question is nonatomic, in the sense
of the definition given in the next section. The aim of this note is to use stable

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics