Reaching Agreement Among $k$ out of $n$ Processes

In agreement problems, each process has an input value and must choose the input of some process (possibly itself) as a decision value. Given $n\geq 2$ processes and $m \geq 2$ possible different input values, we want to design an agreement algorithm that enables as many processes as possible to decide on the same value in the presence of $t$ crash failures. Without communication, when each process simply decides on its input value, at least $\lceil (n-t)/m \rceil$ of the processes are guaranteed to always decide on the same value. Can we do better with communication? For some cases, for example when $m=2$, even in the presence of a single crash failure, the answer is negative in a deterministic asynchronous system where communication is either by using atomic read/write registers or by sending and receiving messages. The answer is positive in other cases.