I am reading now a book about Russian language (The big silver book of Russian language p. 2) and it's written there that when people want to talk to new people then they should say the sentence: "Давайте будем на ты". I've never came across it. Is it indeed done?
ты is the informal usage for you. It is used with family members, children, close friends, and pets. When Russians wish to speak informally to one another for the first time, they say, “ Давайте будем на ты .”
The Russian has two forms for you. They vary according to formality and number.Тыis informal singular, вы is formal singular and both formal and informal plural. Вы is the marker of formality, politeness, and seniority. Whenever you are in doubtabout which form to use, err on the side of caution and use the polite form (Вы) unless you are speaking to a child or a pet.