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In this situation, you are correct that the girlfriend cannot claim your son just because the father gave her permission.  In this case, you should paper file because once you do, the IRS will do an investigation to see who claims the child.  Unfortunately, when that happens your husband can step in and amend his taxes to claim because technically it is his year, but if you do not do the paper file the IRS will just allow the girlfriend to claim because no one disputed it.  Let me know if you have any other questions.
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