On the coverings of the d-cube for d<=6

The cut number S(d) of the d-cube is the minimum number of hyperplanes in R that slice, that is cut the edges while avoiding vertices, all the edges of the d-cube. The cut number problem for the hypercube of dimensions d ≥ 4 was posed by P. O’Neil more than thirty years ago [17]. The identity S(3) = 3 is easy and that of S(4) = 4 is a well-known result, see [5,6,18]. However, the conjecture of S(d) = d is false for d ≥ 6. In fact, M. Paterson found a set of 5 hyperplanes that slice all the edges of the 6-cube, see [18]. Some other new 5-coverings of the 6-cube can be found in [22]. The cut number problem has also been raised by Grünbaum [13], Klee [14] and as a comprehensive approach in Saks [18].