We show that there are CNF formulas which can be refuted in resolution
in both small space and small width, but for which any small-width
proof must have space exceeding by far the linear worst-case upper
bound. This significantly strengthens the space-width trade-offs in
[Ben-Sasson '09]}, and provides one more example of trade-offs in the
"supercritical" regime above worst case recently identified by
[Razborov '16]. We obtain our results by using Razborov's new hardness
condensation technique and combining it with the space lower bounds in
[Ben-Sasson and Nordstrom '08].