Did the mayflower sail with the pinta Nina and santa maria?

No.

The Mayflower was the ship that brought the Puritans (pilgrims) to the new world where they could escape the tyranny of the King of England (James 1) and the Anglican Church. (The church established by Henry the Eighth as the official Church of England (Angle-land) when he broke ties with the pope and started his own church). The Plymouth colony was begun in 1621 by these Pilgrims.

Columbus, 129 years earlier, had sailed in search of a better trade route to the orient. He proposed, properly, that one might sail around the world. He failed, however, when he discovered that the Americas got in the way of that route. His ships were the Pinta, Nina, and Santa Maria.