AWG # 4 copper.
Depends on the size of the service. 100 amp service will require 3 gauge, 150 amp service will require 1/0 gauge, and 200 amp service will require 3/0 gauge.
You listed no gauge wire. This is the required breakers.14 gauge - 15 amp12 gauge - 20 amp10 gauge - 30 amp8 gauge - 40 amp
14 gauge will handle it with a 15 amp breaker. If you use 12 gauge use a 20 amp breaker.
I would recommend you use 1/0 gauge
3 AWG in copper and 2 AWG in Aluminum.
I would run 1/0 gauge.
You listed no gauge wire. This is the required breakers.14 gauge - 15 amp12 gauge - 20 amp10 gauge - 30 amp8 gauge - 40 amp
Number 2 American Wire Gauge
here is a wire gauge calculator which will answer your questions http://www.powerstream.com/Wire_Size.htm
on a 4000 watt amp the best gauge wire woul be between 2 and 0
My memory says correct gauge for 100 amp service is 1 0 (that is, one ought) I could be wrong, check with your power company or building inspector.