8

Since we need to include all 10 printers for any subset $S$, each printer must be connected to fairly enough computers so that it satisfies any situation.

If a printer is connected to less than or equal to 5 computers. There are greater than or equal to 10 computers not connected to it. In this situation, there is a subset $S$ exists that does not match the condition. Therefore, each printer must be connected to at least 6 computers.

Since we have 10 printers, the total number of cables required is

$$6\times 10=\boxed{60}$$