  We provide a number of simplified and improved separations between pairs of
Resolution-with-bounded-conjunction refutation systems, Res(d), as well as
their tree-like versions, Res*(d). The contradictions we use are natural
combinatorial principles: the Least number principle, LNP_n and an ordered
variant thereof, the Induction principle, IP_n.
  LNP_n is known to be easy for Resolution. We prove that its relativisation is
hard for Resolution, and more generally, the relativisation of LNP_n iterated d
times provides a separation between Res(d) and Res(d+1). We prove the same
result for the iterated relativisation of IP_n, where the tree-like variant
Res*(d) is considered instead of Res(d).
  We go on to provide separations between the parameterized versions of Res(1)
and Res(2). Here we are able again to use the relativisation of the LNP_n, but
the classical proof breaks down and we are forced to use an alternative.
Finally, we separate the parameterized versions of Res*(1) and Res*(2). Here,
the relativisation of IP_n will not work as it is, and so we make a vectorising
amendment to it in order to address this shortcoming
