This paper proves exponential separations between depth d-LK and depth (d+1/2)-LK for every d in 0, 1/2, 1, 1 1/2,… utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d-LK and depth (d+1)-LK for d in 0,1,2,… . We investigate the relationship between the sequence-size, tree-size and height of depth d-LK-derivations for d in 0, 1/2, 1, 1 1/2,… and describe transformations between them.
We define a general method to lift principles requiring exponential tree-size (d+1/2)-LK-refutations for d in 0,1,2,… to principles requiring exponential sequence-size d-LK-refutations, which will be described for the Ramsey principle and d=0. From this we also deduce width lower bounds for resolution refutations of the Ramsey principle.