This will keep the flesh right under the pit green, but the rest will still turn brown.
Cut avocados turn brown becaise the flesh reacts with oxygen in the air resulting in the formation of brown pigments such as melanin. Flesh under the pit is sealed away from the air so it stays green, at least for a while. The exposed parts of the flesh will still turn brown. This does not mean the avocado is unsafe to eat. Of course cutting or mashing the avocado just speeds this up.
There are ways that really do work to keep all the flesh from turning brown, at least for while. (1) Press plastic wrap directly onto the surface, (2) Rub with lemon juice, and (3) Store in a small air-tight container with some cut onion. These methods slow the browning but do not delay further ripening/softening.
