I can't get away from the pale turquoise/red combo. You just can't go wrong. And I think the gray would add a really pretty layer to it all. Since it's a basement, I am assuming there isn't a whole lot of natural light so I would keep it pale and bright.

I love the light blue and red (I think it's red..it almost looks orange!) in the "Living in a Bedroom" post. Light blue and red look really good together, and since it's light blue instead of navy, it won't look like a "patriotic" room. You could also get a gray or white chair to go in the room to add a little contrast without competing with bright walls.

I agree with the blues - I think there are several light blues that could look great.

I also think it might be worth looking at some yellows. Without being in the space I would hesitate recommending a specific tone - but I think yellow walls with a red sofa can look really good if done carefully.

I'm wondering thought how the yellow walls would look with the grey carpet. Are you thinking about very pale yellows.

My living room (w/dark red and dark gray furniture/accents) is painted lavender. I actually expected it to look more blue and wouldn't have chosen such a lavender shade, but now that it's there, I love it. It's bright and cheery.

But I also like the red/pale turquoise combination.

How about painting the walls warm buttery white and accent with turquoise, golds and orange.

Here are some combos of turquoise, reds-
http://www.houseofturquoise.com/search/label/Turquoise%20and%20Red

I would paint the walls white so you can add any art that you want. For the chair I would choose neutral or white.

A grayed-down golden yellow (see West Elm's current "yellow," Turmeric, for a reference if you want something upbeat.)

If you want cozy and chic (and have decent installed lighting), match the walls as closely as possible to the carpet.

Otherwise, white... and an accent wall of either the yellow or the red... or wide horizontal stripes of both!