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Wow, someone actually asking BEFORE they do it. That is very exciting, people don't often do it with respect to run fencing! Good for you
It depends on how large your run will be (very small runs, like 8x8 for instance, can get away with weeblier fences and longer post spacing than 'real' sized runs), how windy your site is and how squishy or sandy your soil can be (the windier the site or the softer the soil, the more posts you should use and the deeper you should set them) and how likely you think it is that the fence will be *seriously* challenged by large predators, or for that matter backed into by the lawnmower.
With 4x4 posts and a "normal to large" run and "normal" soil (whatever normal is
) I would say pick a number between 5-8 feet, certainly no further apart than 8 ft. With high wind, soft soil or high risk, use more posts.
Btw, if by "poultry wire" you mean hex chickenwire, be aware that the great majority of what's sold today is BY NO MEANS predatorproof. It just keeps chickens from wandering out. I would very very strongly suggest something, well, stronger, like good quality 1x1 welded wire, or good quality 2x4 welded wire with something smaller-gauge added along the bottom. Be careful of anything 2x2 or 2x3 which IME tends to be extremely weebly and light-duty -- small-gauge and crappily welded, I would not use it for predatorproofing.
You will also want some means of keeping predators from digging in, I don't know if you've planned this already but a wire apron at least 2' wide is a good idea all around the run.
Good luck, have fun,
Pat