New York Laws
Article 10 - Parties Generally
R1009 - Claim by Plaintiff Against Third-Party Defendant.

Rule 1009. Claim by plaintiff against third-party defendant. Within
twenty days after service of the answer to the third-party complaint
upon plaintiff's attorney, the plaintiff may amend his complaint without
leave of court to assert against the third-party defendant any claim
plaintiff has against the third-party defendant.