Density

Let X be a finite set. The usual definition of the density of a subset Y of X is |Y|/|X|, that is, the size of Y divided by the size of X. In particular, if $\mathcal{A}$ is a subset of $[3]^n$ then its density is $3^{-n}|\mathcal{A}|.$
One speaks loosely of a set $\mathcal{A}\subset[3]^n$ being dense if its density $\delta$ is bounded below by a positive constant that is independent of n. Strictly speaking, this definition applies to sequences of sets with n tending to infinity, but it is a very useful way of talking.
Sometimes it is helpful to consider other probability measures on $[3]^n,$ such as equal-slices density. Then the words "density" and "dense" have obviously analogous uses.