If a custody matter has not yet been filed, then under Oregon law, the parent with whom the child was living at the time paternity of the child was determined is determined to have legal custody of the child. ORS 109.175 provides: (1) If paternity of a child born out of wedlock is established pursuant to a petition filed under ORS 109.125 or an order or judgment entered pursuant to ORS 109.124 to 109.230 or ORS 416.400 to 416.465, or if paternity is established by the filing of a voluntary acknowledgment of paternity as provided by ORS 109.070 (1)(e), the parent with physical custody at the time of filing of the petition or the notice under ORS 416.415, or the parent with physical custody at the time of the filing of the voluntary acknowledgment of paternity, has sole legal custody until a court specifically orders otherwise.
If paternity was not established, then the father does not have any legal rights to the child until paternity is established.
The best way to address this issue is to file a petition for custody and obtain a status quo order pursuant to ORS 107.097 which will maintain the child’s “usual place of residence,” “present placement and daily schedule,”party’s usual contact and parenting time,” and “current schedule and daily routine of the child.” This will prevent the other parent from taking the child and running. You should also do your best to document the other parent’s statements.