  Results of Sierpinski and others have shown that certain finite-dimensional
product sets can be written as unions of subsets, each of which is "narrow" in
a corresponding direction; that is, each line in that direction intersects the
subset in a small set. For example, if the set (omega \times omega) is
partitioned into two pieces along the diagonal, then one piece meets every
horizontal line in a finite set, and the other piece meets each vertical line
in a finite set. Such partitions or coverings can exist only when the sets
forming the product are of limited size.
  This paper considers such coverings for products of infinitely many sets
(usually a product of omega copies of the same cardinal kappa). In this case, a
covering of the product by narrow sets, one for each coordinate direction, will
exist no matter how large the factor sets are. But if one restricts the sets
used in the covering (for instance, requiring them to be Borel in a product
topology), then the existence of narrow coverings is related to a number of
large cardinal properties: partition cardinals, the free subset problem,
nonregular ultrafilters, and so on.
  One result given here is a relative consistency proof for a hypothesis used
by S. Mrowka to construct a counterexample in the dimension theory of metric
spaces.
