Gauge is the thickness of the wire. The wire spacing is a different issue.
I use 19 gauge vinyl coated galvanized after weld wire, 1/2" by 1/2" wire spacing. But we don't have coyotes or fox in our fenced suburban back yard...at least I hope not!
I've read about some people using a lower gauge (that means it's thicker) wire for more strength: say, 16 or 14 gauge. But it's a lot more expensive to get 1/2" by 1/2" wire in these gauges. If you use wire with greater spacing, you've got to consider the possibility that predators like raccoons could reach through the wire, grab your bird and pull pieces of it through the wire. I've read about this happening even with wire spacing of 1/2" by 1".
You could use a lower gauge wire with greater wire spacing for your run, then reinforce the lower two or three feet (and anywhere perches come up close to the wire) with higher gauge but 1/2" by 1/2" inch spaced wire. That way you get the best of both approaches.