To put it bluntly, yes. Lorelai and Luke did get together for 4 months. Then they broke up due to Emily. Then they got back together. I'd go into more detail but I've already answered the question.
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Alas, no they did not stay together because Lorelai gave Luke an ultimatum and then went and slept with Christopher when he said no. Lorelai married Christopher, but only for a short time because she realized that even without Luke, Christopher wasn't the one. Luke and Lorelai did get back together in Bon Voyage, but that is when the show ended.
Lauren Graham plays Lorelai Gilmore. Alexis Bledel plays Lorelai 'Rory' Gilmore.
June 3rd was the day Lorelai and Luke were supposed to get married until the day Lorelai found out about April, Luke's 12 year old daughter. Luke said it was too much all at once and they decided to postpone, that being the beginning of the end of their relationship.
Luke and Lorelai get engaged, but Luke postpones the wedding after finding out that he has a daughter (April Nardini). Tired of waiting for Luke, Lorelai gives him an ultimatum: he must elope with her immediately or not marry her at all. When he refuses to elope, Lorelai immediately visits Christopher. Two days later, Luke is ready to elope, but Lorelai tells him about her night with Christopher and Luke and Lorelai break up. Lorelai and Christopher get married but soon decide to get divorced. At the end of the seventh (final) season, Luke and Lorelai get back together. However, they are never married on the show.
No and there will never be that show has ended and it is so 2 years ago
The Gilmore girls cast list were - Lauren Graham - Lorelai Gilmore Alexis Bledel - Rory Gilmore Keiko Agena - Lane Kim Scott Patterson - Luke Danes Yanic Truesdale - Michel Gerard Kelly Bishop - Emily Gilmore Edward Herman - Richard Gilmore Melissa Mc.Carthy - Sookie St.James Sean Gunn - Kirk Gleason Liza Weil - Paris Geller Liz Torres - Miss Patty Jared Padalecki - Dean Forester Matt Czuchry - Logan Huntszberger Jackson Douglas - Jackson Belleville Michael Winters - Taylor Doose Sally Struthers - Babette Dell Emily Kuroda - Mrs.Kim Todd Lowe - Zack Van gerbig Milo Ventimiglia - Jess Mariano David Sutcliffe - Christoper Hayden Teal Redmann - Louise Grant Shelly Cole - Madeline Lynn] and many many more...