  In the context of proving lower bounds on proof space in k-DNF resolution,
[Ben-Sasson and Nordstrom 2009] introduced the concept of minimally
unsatisfiable sets of k-DNF formulas and proved that a minimally unsatisfiable
k-DNF set with m formulas can have at most O((mk)^(k+1)) variables. They also
gave an example of such sets with Omega(mk^2) variables.
  In this paper we significantly improve the lower bound to Omega(m)^k, which
almost matches the upper bound above. Furthermore, we show that this implies
that the analysis of their technique for proving time-space separations and
trade-offs for k-DNF resolution is almost tight. This means that although it is
possible, or even plausible, that stronger results than in [Ben-Sasson and
Nordstrom 2009] should hold, a fundamentally different approach would be needed
to obtain such results.
