N.Y. C.P.L.R. § 1007
After the service of his answer, a defendant may proceed against a person not a party who is or may be liable to that defendant for all or part of the plaintiff's claim against that defendant, by filing pursuant to section three hundred four of this chapter a third-party summons and complaint with the clerk of the court in the county in which the main action is pending, for which a separate index number shall not be issued but a separate index number fee shall be collected. The third-party summons and complaint and all prior pleadings served in the action shall be served upon such person within one hundred twenty days of the filing. A defendant serving a third-party complaint shall be styled a third-party plaintiff and the person so served shall be styled a third-party defendant. The defendant shall also serve a copy of such third-party complaint upon the plaintiff's attorney simultaneously upon issuance for service of the third-party complaint on the third-party defendant.
N.Y. CPLR 1007