My son lives with me, in my house, full time except for when he is away at college. I pay 100% percent of the costs for this house. He is 19 years old and a full time student. The divorce agreement states that my ex wife and I will alternate years in which we claim our son as a dependent but is silent on head of household.
Under the special rules for children of divorced or separated parents, only the parent where the child lives more than half the nights of the year can use the child as a qualifying person for head of household status. If this is your ex's year to claim the child as a dependent, you must give your ex a signed copy of form 8332, which you can find under "other tax situations" or download separately from the IRS web site. Your ex can claim the dependent exemption and the child tax credit (except your child has aged out of that) but the claim for HOH and also claims for certain tuition benefits always stay with the custodial parent.
See here
Publication 501, Exemptions, Standard Deduction, and Filing - IRS.gov
Under the special rules for children of divorced or separated parents, only the parent where the child lives more than half the nights of the year can use the child as a qualifying person for head of household status. If this is your ex's year to claim the child as a dependent, you must give your ex a signed copy of form 8332, which you can find under "other tax situations" or download separately from the IRS web site. Your ex can claim the dependent exemption and the child tax credit (except your child has aged out of that) but the claim for HOH and also claims for certain tuition benefits always stay with the custodial parent.
See here
Publication 501, Exemptions, Standard Deduction, and Filing - IRS.gov
Yes, you claim Head of Household sine the child lives with you. You can also claim the Earned Income Credit. Be sure the ex knows you are doing so to avoid a potential conflict.
"The divorce agreement is silent on head of household." It should be. The courts can not grant a tax benefit not allowed by law.
No, you cannot file as HOH unless you are claiming a blood-related dependent.