If we don't use any concrete, and only sink the posts about 18 inches, will it be reasonably stable?

We have rocky clay soil and sometimes a pick ax is required to dig deep holes so go easy on us with your advice. lol We may have to start using the pick ax at around 15 to 18 inches deep. It's hard to know until we start digging.

I will be planting a lonicera sempervirens on each side.

However, you still need to consider frost heave that could cause part of your arbor to shift, so should drop your posts deep enough to get below your frost line, which in zone 6, shouldn't be below 30 inches. Dig down at least 30 inches, fill the bottom 6" with gravel, then set your posts and back fill with soil, packing it well.

BUT- with a 5 X 2- 1/2 ft arbor- why bother setting the corner posts into the ground at all?

My arbor is nearly the same size and we mounted the corner posts to some treated 4 X 4's that serve as skids, laying on the ground. We drilled holes and locked them down with rebar. Eventually, those short pieces of 4 x 4 base may rot, but the arbor itself has no direct ground contact, and if ever necessary replacing those 2 base pieces will be easy. It will also allow me to move it if necessary, for maintenance or reconfiguration of the space.

DH used this technique for my clematis trellises in the photo below. They are eight feet tall in the middle and seven feet wide and very stable.

I do it the easy way. First I dig a deep hole. with the post in the hole, then I fill the hole up with bagged concrete mix (dry). Then I just add water to the concrete mix and let the mix cure for a few days. No mixing at all.

I have set up msny posts that way.

dcarch

We used pieces of 4 x 4's because we had them. They were cut offs from the 8 ft. ones we used as corner posts, but 2 x 4's would work just as well.

The entire purpose is to have EITHER a floating OR footed foundation, but NOT a bit of both. If it's footed, the footers must be deep enough to totally avoid frost heave. Otherwise, it's better to just let it float, because any footer that moves in the least bit while others remain firm will torque the structure.

That's why I like using rebar in pre-drilled holes. It will hold the 4 x 4 or 2 x 4 anchor pieces down securely, but if there is any heaving, the pieces of rebar may pop up a wee bit, or sink down, but in either case, a few taps with a mallet puts it all back to rights quickly and easily, but the structure to which they are secured has moved with them, so remains unaffected.

Hope that made sense.