Valachi says that the phrase "to go the mattresses" originated during an underworld upheaval in the early 1930s known as the Castellamerese War. Gangs would rent empty buildings and install assassins in them to target members of opposing factions. Since this often involved days or weeks of surveillance, mattresses were needed for the murder teams to sleep on until the "job" was done. The phrase "to go to the mattresses" naturally came to acquire the meaning of starting a gang war - the context in which it is used in The Godfather.
Valachi is also responsible for revealing the name that most New York members of the "Mafia" gave to their organisation - Cosa Nostra.