Let us call $f,g:\omega\to \omega$ *almost totally distinct* if $$|\{n\in \omega: f(n) = g(n)\}| < \aleph\_0.\;\;\;\; (\star)$$
It is known that there are uncountable collections of almost totally distinct functions. 


**Question.** Is the above statement still true if we replace $\aleph\_0$ by $2$ in $(\star)$?



Almost totally distinct functions
No. By pigeonhole principle any uncountable family of functions $f:\omega\to \omega$ contains two functions with the same pair $(f(1),f(2))$.


