Facebook SDK 3.1 for iOS - runs on iOS6, but crashes on iOS 5.x

Did you set the frameworks to be optional? When you are adding AdSupport.framework, Social.framework, and Accounts.framework, there is drop down menu to the right that you can select between "Required" and "Optional". See a picture example here: Link

Another thing to check is in your Project's "Build Settings" that 'Base SDK' is 6.0 and 'iOS Deployment Target' is iOS 4.3.

I'm able to build FB SDK 3.1 on my iOS 5.1 with these settings.

If you're targeting iOS versions less than 6.0, you'll need to make AdSupport.framework, Social.framework, and Accounts.framework optionally-linked.

Please have a look at the attached screenshot.

Cheers...!!!