id: 05175710
dt: a
an: 05175710
au: Gao, Su; Miller, Arnold W.; Weiss, William A.R.
ti: Steinhaus sets and Jackson sets.
so: Gao, Su (ed.) et al., Advances in logic. The North Texas logic conference,
    Denton, TX, USA, October 8‒10, 2004. Providence, RI: American
    Mathematical Society (AMS) (ISBN 978-0-8218-3819-8/pbk). Contemporary
    Mathematics 425, 127-145 (2007).
py: 2007
pu: Providence, RI: American Mathematical Society (AMS)
la: EN
cc: 
ut: Jackson set; Steinhaus problem
ci: Zbl 1070.52008
li: 
ab: A finite set $X\subseteq\mathbb R^{2}$ is {\it Jackson} if for every set
    $S\subseteq\mathbb R^{2}$ there is an isometric copy $Y$ of $X$ with
    $\vert Y\cap S\vert \ne 1$.  The central problem for Jackson sets is
    the following: is every finite set with two or more points Jackson.
    This problem has been raised by Jackson in the context of the solution
    of a problem of Steinhaus [see e.g. {\it S. Jackson} and {\it R. D.
    Mauldin}, Bull. Symb. Log. 9, 335‒361 (2003; Zbl 1070.52008)].
    Jackson proved that every set of three points in $\mathbb R^{2}$ is
    Jackson.  In the present article the authors give several contributions
    to this problem including the following interesting result: The set of
    vertices of the unit square is Jackson.  The article closes with
    several open questions.
rv: Peter M. Gruber (Wien)