| Jumper the R and w terminal at the furnace. If the furnace doesn't come on and stay on until you remove the jumper, the problem is in the furnace. If it still cuts out, you need to observe what happens at the time. Does the power on light go out? Does the blower of gas go off first, etc.?
If it stays on, and was doing the same thing with a new thermostat, I would look at the wiring very carefully. | 
Linear Mode
