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On infinite partitions of lines and space

100%

Erdös P., Jackson S., Daniel Mauldin R.

Fundamenta Mathematicae

1997

tom 152

nr 1

75-95

Given a partition P:L → ω of the lines in $ℝ^n$, n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, $Q:ℝ^n → ω$, so that each line meets at most m points of its color. Assuming Martin's Axiom, we show this is the case for m ≥ 3. We reduce the problem for m = 2 to a purely finitary geometry problem. Although we have established a very similar, but somewhat simpler, version of the geometry conjecture, we leave the general problem open. We consider also various generalizations of these results, including to higher dimension spaces and planes.

Almost disjoint families and property (a)

58%

Szeptycki P. J., Vaughan J. E.

Fundamenta Mathematicae

1998

tom 158

nr 3

229-240

We consider the question: when does a Ψ-space satisfy property (a)? We show that if $|A| < \got p$ then the Ψ-space Ψ(A) satisfies property (a), but in some Cohen models the negation of CH holds and every uncountable Ψ-space fails to satisfy property (a). We also show that in a model of Fleissner and Miller there exists a Ψ-space of cardinality $\got p$ which has property (a). We extend a theorem of Matveev relating the existence of certain closed discrete subsets with the failure of property (a).

Ograniczanie wyników

2 Fundamenta Mathematicae

1 Daniel Mauldin R.

1 Erdös P.

1 Jackson S.

1 Szeptycki P. J.

1 Vaughan J. E.

1 1998

1 1997

Wyniki wyszukiwania

On infinite partitions of lines and space

Almost disjoint families and property (a)