Connected Hausdorff spaces with large collection of disjoint open sets Is there for every infinite cardinal $\kappa$ a connected Hausdorff space $(X,\tau)$ with $|X| = \kappa$ and a collection ${\cal D}$ of mutually disjoint open sets with $|{\cal D}| = \kappa$?
 A: For every $\ \kappa\ge|\Bbb R| \ $ the cone
$\ X=\text{cone(D)}\ $ over the discrete space $\ D\ $ of cardinality $\ \kappa\ $ is an example.
The required open sets are the cone lines minus the vertex.
When it comes to lower cardinalities then I'd look into the Urysohn countable connected Hausdorff space.
In particular, let's apply the Bing's example of Hausdorff connected countable space, $\ B\ $ (see https://www.ams.org/journals/proc/1953-004-03/S0002-9939-1953-0060806-9/S0002-9939-1953-0060806-9.pdf)
Any infinite family of pairwise disjoint euclidean-open subsets of $\ \Bbb Q\times\{0\}\ $ are open also in $\ B.\ $
Thus the OP's problem is solved for all cardinalities but the weird ones, between $\ |\Bbb Q|\ $ and $\ |\Bbb R|,\ $ that exist only when we don't accept the continuum hypothesis.
A: Based on the previous answer of Wlod AA, such a space always exists for $\kappa$ infinite (it is clear that it cannot exist for $2\leq\kappa<\omega$).
Let $I$ be a countable Hausdorff connected space (see A connected countable Hausdorff space, R. H. Bing, 1952), and $x$ any point in $I$. Then the quotient $(I\times\kappa)/(\lbrace x\rbrace\times\kappa)$ is such a space ($\kappa$ is considered as a discrete space), with $\mathcal D$ the collection of open sets $(I\setminus\lbrace x\rbrace)\times\lbrace\alpha\rbrace$ for $\alpha<\kappa$.
