Three and four variables configurations Karnaugh maps will probably be the usual representations of data. One can also easily imagine five- and six-variable configurations. For configurations of 5 variables, one could just put side by side two 4-variable Karnaugh maps, where the left one is for v5=0 and the right one v5=1. The six-variable scheme would be to replicate the latter 5-variable Karnaugh map, on two "meta" rows: one meta-row would be the two previous 5-variable Karnaugh maps, for which v6=0; the second meta-row would represent cases of v6=1.
Anyone with a good vision for three-dimensional perspective would rather extends Karnaugh maps with stacked layers, instead of put them side by side. As an illustrative case, one could repeat the rows and column structure 00, 01, 11 and 10 on a vertical axis. This would provide the necessary space for a 6-variable Karnaugh map. No doubt this sounds exciting, but may not be that easy to understand. So for the moment we restrain ourselves to 3- and 4-variable configurations.