Answer: It sounds lovely! I'm going to presume this is in full sun, as you don't mention any shade.
If you want to complete the patriotic palette, you can grow a red morning glory or cardinal creeper on the arbor. Both are annual vines that grow vigorously and bloom all summer long. Morning glories also come in an array of other colors (from white to deep purple-black), so you could even grow a rainbow of them on your arbor. Another quick grower that will provide you with an edible reward is scarlet runner bean.
If you're more interested in a perennial vine, a honeysuckle (Lonicera x heckrottii) or trumpet creeper (Campsis radicans) would cover the arbor quickly and also provide summer-long bloom that hummingbirds love. Both require pruning to maintain their shape and to prevent them from spreading too far. Lonicera should be pruned after it finishes flowering with heading cuts to promote thick growth. C. radicans needs pruning in early spring before growth begins. Either can be trimmed at any time during the growing season to tame rampant growth.
There are lots of other vines to consider, but these would fit well, I think. Enjoy!
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