I'm not a lawyer, but if the son lives with his mother and they both live with the son's grandmother (mother's mother), and the grandmother provides over 1/2 the son's support, the grandmother could claim the child. If the child does not live with the grandmother over 1/2 the year in the US, the grandmother wouldn't be able to claim him for EITC. In my tax office, we ask for proof that the child lives at the address of the taxpayer claiming the child. Proof would include a doctor's or school record with the correct address on it. Anyone who claims a deduction or credit that they are not entitled to could be looked at as fraud by the IRS. Making false claims to get EITC, if caught, could result in loss of eligibility for the EITC in the future.