This is a reformulation of this MO question which recieved little or no attention due to the fact that the OP gave no motivation whatsoever. I found the question quite interesting and decided to give it another try (if this is not OK please let me know and I´ll delete this post).

It is known that if $G$ is an abelian compact topological group then it contains a dense subgroup $H$ which is countably tight (in fact Frechet-Urysohn). However the following is open (at least it was a few years ago):

If $G$ is a compact group, must $G$ contain a dense subspace of countable tightness?

This is problem 4.1.1 in "Topological Groups and Related Structures" by A.Arhangelskii and M.Thachenko. Problem 4.1.7 (also open) in the same book is:

Is it true that every homogeneous compact space contains a dense subspace of countable tightness?

My guess is that there should be known examples of (non-homogeneous) compact spaces such that any dense subspace has uncountable tightness, but I could not find any. So I have two questions:

1. Is there such a compact space?

2. For a cardinal $\kappa > 2^{\aleph_0}$ what is a dense subspace of $[0,1]^\kappa$ that has countable tightness?

Perhaps the answer to 2) is that there is none, and $[0,1]^\kappa$ is indeed a counterexample for the second quoted question, but I wouldn´t expect that. Note that if $\kappa \leq 2^{\aleph_0}$ then $[0,1]^\kappa$ is separable and any countable dense subspace would do the trick.