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This is why paint chips exist. Go get some! When I think of what color goes with an existing item, I mentally go across the color wheel for the complementary color. You can imagine a green--remember there are a million sorts of green--which is the color across from red. This is a complementary color scheme, and it can work well.
If you go get a bunch of green chips, you can probably visualize which sort of green will look best. If your spread is somewhat grayed, then probably a grayed green will work. Also remember that gray is having a moment now, and if you are lucky enough to have bright white woodworks, any sort of gray, including a green gray or gray green.
If you can find a really good color wheel, you can even see which color is opposite which of the reds you have in your bedroom.
Another method for determining wall color is to take an item as your inspiration piece. It sounds as if your bedding is that inspiration piece for you. Another poster mentioned using the gold in your bedding as a possibility for your walls, and I agree. Don't try to match, but try to find a color that seems right. I think you want to steer away from too deep a color--probably. Its hard to say from the details you've given.
Right now, wall colors are often various permutations of grayed neutrals. Using such a color is on trend right now.
In terms of design, 3 colors are typically used: a primary, secondary, and tertiary (Primary being the most dominant and so forth). Sounds like you've already got the three colors: burgundy, gold, ivory. So to choose something in the burgundy or gold family may work out.
Another idea: If you're not sure about the color, buy a quart and just paint one wall. Live with it for awhile, like a few weeks and see if you really LOVE it. That small paint chip/sample can be deceiving when thinking of such a large space.
I would definitely not use blue. You've got warm colors, blue is a cool color. Plus blue in a bedroom is depressing (think Picasso and his Blue Period). Reds are warm, cozy, and definitely more romantic although too much red can be overwhelming and may cause restlessness.
I know this thread is old and I'm sure the OP has painted her walls by now, but I just had to chime in on the comment about blue being depressing in a bedroom. I completely disagree. I have my bedroom painted in a Sherwin-Williams color called "Rain" that is so soothing, I feel like I'm in a spa. This color has enough blue in it to not be dreary (too grayish), but not so much blue that it looks like a baby's room.