Is the Gilbert family history being said in the show real?

In one episode (1.09) it is said that a Gilbert was on the Mayflower. I goggled the Mayflower and Gilbert, and a 인기 name that came up was John Gilbert (Elena's dads name). Although it did not say anything about how he was related to the Mayflower I would still like to know how and if this man is who they were talking about in that episode when they mentioned the Mayflower. Please help because I am a real Gilbert myself and would 사랑 to know some family history, and if I'm related to the fictional Vampire Diaries characters!!! :)