Perfectly meager sets and universally null sets

We will show that there is no example of a set distinguishing between universally null and perfectly meager sets.

     

Remarks on small sets of reals

We show that the Dual Borel Conjecture implies that \boldsymbol\aleph_1 $"> and find some topological characterizations of perfectly meager and universally meager sets.

     

Strongly meager sets of size continuum

We will construct several models where there are no strongly meager sets of size 2⁰. First author partially supported by NSF grant DMS 0200671.Second author partially supported by Israel Science Foundation and NSF grant DMS 0072560. Publication 807. Mathematics Subject Classification (2000):03E15, 03E20     

On perfectly meager sets in the transitive sense

We prove that assuming one can always find a perfectly meager set, which is not perfectly meager in the transitive sense.

     

Strongly meager sets and their uniformly continuous images

We prove the following theorems:

(1) Suppose that is a continuous function and is a Sierpinski set. Then

(A)

for any strongly measure zero set , the image is an -set,

(B)

is a perfectly meager set in the transitive sense.

(2) Every strongly meager set is completely Ramsey null.

     

Strongly meager sets of real numbers and tree forcing notions

We show that every strongly meager set has the - and the - property.

     

Universally meager sets

We study category counterparts of the notion of a universal measure zero set of reals.

We say that a set is universally meager if every Borel isomorphic image of is meager in . We give various equivalent definitions emphasizing analogies with the universally null sets of reals.

In particular, two problems emerging from an earlier work of Grzegorek are solved.

     

On a conjecture of Hoàng and Tu concerning perfectly orderable graphs

Hoàng and Tu [On the perfect orderability of unions of two graphs, J. Graph Theory 33 (2000) 32-43] conjectured that a weakly triangulated graph which does not contain a chordless path with six vertices is perfectly orderable. We present a counter example to this conjecture.     

Universally meager sets, II

We present new characterizations of universally meager sets, shown in [P. Zakrzewski, Universally meager sets, Proc. Amer. Math. Soc. 129 (6) (2001) 1793-1798] to be a category analog of universally null sets. In particular, we address the question of how this class is related to another class of universally meager sets, recently introduced by Todorcevic [S. Todorcevic, Universally meager sets and principles of generic continuity and selection in Banach spaces, Adv. Math. 208 (2007) 274-298].