  NOTE: Unfortunately, most of the results mentioned here were already known
under the name of "d-separated interval piercing". The result that T_d(m)
exists was first proved by Gya\'rfa\'s and Lehel in 1970, see [5]. Later, the
result was strengthened by Ka\'rolyi and Tardos [9] to match our result.
Moreover, their proof (in a different notation, of course) uses ideas very
similar to ours and leads to a similar recurrence. Also, our conjecture turns
out to be right and was proved for the 2-dimensional case by Tardos and for the
general case by Kaiser [8]. An excellent survey article ("Transversals of
d-intervals') is available on http://www.renyi.hu/~tardos. Still, we leave this
paper available to the public on http://page.mi.fu-berlin.de/dawerner, also
because one might find the references useful.
  -----
  We study the following Gallai-type of problem: Assume that we are given a
family X of convex objects in R^d such that among any subset of size m, there
is an axis-parallel hyperplane intersecting at least two of the objects. What
can we say about the number of axis-parallel hyperplanes that sufficient to
intersect all sets in the family?
  In this paper, we show that this number T_d(m) exists, i.e., depends only on
m and the dimension d, but not on the size of the set X. First, we derive a
very weak super-exponential bound. Using this result, by a simple proof we are
able to show that this number is even polynomially bounded for any fixed d.
  We partly answer open problem 74 on http://maven.smith.edu/~orourke/TOPP/,
where the planar case is considered, by improving the best known exponential
bound to O(m^2).
