**
A big symmetric planar set with small category projections
**

by

**
Krzysztof Ciesielski
and
Tomasz Natkaniec
**

*Fund. Math. 178(3)* (2003), 237-253.

We show that under appropriate set theoretic assumptions
(which follow from Martin's axiom and the continuum hypothesis)
there exists a nowhere meager subset
A of **R** such that

- for each continuous nowhere constant function
f:
**R**-->**R** the set
{c in **R**: proj[(f+c)\cap (AxA)] is not meager}
is meager, and
- for each continuous f:
**R**-->**R** the set
{c in **R**: (f+c)\cap (AxA) is empty}
is nowhere meager.

The existence of such a set follows also from the principle CPA,
which holds in the iterated perfect set model.
We also prove that the existence of a set A as in (1)
cannot be proved in ZFC alone even when we restrict our attention to
homeomorphisms
of **R**. On the other hand, for the class of real analytic functions
a Bernstein set A satisfying (2) exists in ZFC.

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**Last modified October 24, 2003.**