Is there a cardinal $\kappa>2^\omega$ and a connected space $X$ such that 


(1) $|X|=\kappa$, and 


(2) every connected subset of $X$ (with at least 2 points) has cardinal $\kappa$? 


Let's assume every space is regular, so the answer would be YES if we allowed $\kappa=2^\omega$.


I seem to recall a paper with an example, but that could be wrong. 



Cardinality of connected subspaces
The answer is yes. Consider any limit ordinal $\gamma$ and let
$X$ be the space of all binary $\gamma$-sequences that are not
eventually ones, ordered under the lexical ordering. We may put the
order topology on $X$. This space has size $\kappa=2^\gamma$.


I claim that this space, in the order topology, is
connected. To see this, note first that every bounded set in $X$
has a least upper bound. This is just like some of the usual proof that the
reals are complete. Combining with the fact that the order is dense
implies connectedness.


Next, every nontrivial open interval in the space has size $2^\gamma$, because
it must contain an interval, which has size $2^\gamma$, which is
$\kappa$.


So every connected subspace with at least two points has size $\kappa$, as desired. By choosing $\gamma$ large enough, we can ensure $\kappa=2^\gamma>2^\omega$. 


