Minimal subcoverings of a cover whose sets intersect in at most $1$ point Suppose we are given a nonempty set $X$. Let ${\cal U}$ be a set of subsets of $X$ such that


*

*$\bigcup {\cal U} = X$, and

*$U_1\neq U_2\in {\cal U}$ implies $|U_1\cap U_2| \leq 1$.


Is there a subcollection ${\cal U}_0\subseteq {\cal U}$ such that


*

*$\bigcup {\cal U}_0 = X$, and

*if $U\in{\cal U}_0$ then $\bigcup \big({\cal U}_0 \setminus \{U\}\big) \ne X$
?

 A: Edit 08/21/2017
Péter Komjáth detected a gap in my original argument, so I will edit it now to limit what is claimed.
I claim that if (i) $|X|=\omega$, or if (ii) $|X|=\kappa>\omega$ and $\mathcal U$ is a locally finite cover of $X$, then the question has an affirmative answer. Here I say that $\mathcal U$ is a locally finite cover of $X$ if each element of $X$ is contained in only finitely many elements of $\mathcal U$. 
From this point until the last paragraph, the only edits I intend to make involve replacing some $U$'s with ${\mathcal U}$'s. (These were typos in my original writeup.)
I will explain how to construct ${\mathcal U}_0$.
Let $(x_{\lambda})_{\lambda<\kappa}$ be an enumeration of $X$.
I will examine the elements of $X$, roughly in order,
and decide which elements
of ${\mathcal U}$ to put into ${\mathcal U}_0$.
To keep track of things,
I will write $(X,\emptyset)$ and $({\mathcal U},\emptyset)$
to indicate
the starting state and $(\emptyset,X)$ and $(\emptyset, {\mathcal U}_0)$
to indicate the ending state. Middle states will be of the form
$(X',X'')$ and $({\mathcal U}', {\mathcal U}'')$
where $\{X',X''\}$ is a partition of $X$ into two cells,
where the elements of $X''$ have been `handled'
and the elements of $X'$ have yet to be handled.
As we handle the elements of $X$ we move them
from $X'$ to $X''$ and either discard
sets from ${\mathcal U}'$ or else we move some sets
from ${\mathcal U}'$ to ${\mathcal U}''$.
Throughout the process we will arrange that $X'$
is contained in the union of the sets in ${\mathcal U}'$,
while ${\mathcal U}''$
is a minimal cover of $X''$.
The process will end with a minimal
cover ${\mathcal U}_0$ of $X$.
As I move through the process
I will want to maintain an additional
assumption. As mentioned above, 
I will assume at each stage 
$(X',X'')$ and $({\mathcal U}', {\mathcal U}'')$
that $X'$
is contained in the union of the sets in ${\mathcal U}'$,
and ${\mathcal U}''$
is a minimal cover of $X''$.
(This includes the assumption that $\bigcup_{u\in {\mathcal U}''}u=X''$.)
For each $v\in {\mathcal U}''$ the set
${\mathcal U}''-\{v\}$
does not cover $X''$, so there is an element
$x_v\in X''$ contained in $v$ and in no other
element of ${\mathcal U}''$. Call $x_v$ an anchor
for $v$. The assumption that ${\mathcal U}''$ is a minimal
cover of $X''$ means exactly that every $v\in {\mathcal U}''$
has an anchor in $X''$. The additional 
assumption I intend to maintain throughout the process is:
no set in ${\mathcal U}'$ contains
an anchor $x_v\in X''$ of an element $v\in {\mathcal U}''$.
The additional assumption holds in the starting state,
since ${\mathcal U}''=\emptyset$ at the beginning.
The construction really starts now.
I allow two ways for the process to move forward.
A move of type 1.
We are at stage $(X',X'')$ and $({\mathcal U}', {\mathcal U}'')$,
where (i) $X'$ is contained in the union of the sets
in ${\mathcal U}'$, (ii) ${\mathcal U}''$ is a minimal cover of 
$X''$, and (iii) no set in ${\mathcal U}'$
contains an anchor $x_v\in X''$ of an element $v\in {\mathcal U}''$.
I define a directed graph with vertex set $X'$.
Write $x\to y$ if the implication
[$x\in u$ implies $y\in u$] holds
for every $u\in {\mathcal U}'$.
There will be loops on every $x\in X'$, but they are unimportant.
If $x\to y$ is a nonloop, then any $u\in {\mathcal U}'$
that contains $x$ must also contain $y$. Therefore there cannot
be two sets $u, v\in {\mathcal U}'$ containing $x$,
since this would yield $u\cap v \supseteq \{x,y\}$,
contradicting $|u\cap v|\leq 1$. Thus each $x\in X'$
that is the tail vertex of a nonloop is contained
in a unique $u_x\in {\mathcal U}'$.
I intend to (i) move all tail
vertices $x$ of nonloops from $X'$ to $X''$,
move all associated sets $u_x$
from ${\mathcal U}'$ to ${\mathcal U}''$,
(iii) treat $x$ as the anchor of $u_x$,
and (iv) move from $X'$ to $X''$
all other elements covered by the $u_x$'s.
I claim that after these steps
we again have a stage
$(\overline{X}',\overline{X}'')$ and
$(\overline{\mathcal U}', \overline{\mathcal U}'')$,
where the assumptions are satisfied.
(I overlined sets $X', X'', {\mathcal U}', {\mathcal U}''$
to indicate
the state of the set after the step is completed.)
To see that $\overline{X}'\subseteq
\bigcup_{u\in \overline{\mathcal U}'} u$  it suffices
to note that $\overline{X}'\cup \overline{X}''=X= {X}'\cup {X}''$,
$\overline{\mathcal U}'\cup \overline{\mathcal U}''= {\mathcal U}'\cup {\mathcal U}''$ (which covers $X$),
and $\overline{X}''=\bigcup_{u\in \overline{\mathcal U}''} u$.
To see that $\overline{\mathcal U}''$ is a minimal cover of $\overline{X}''$
it suffices to note that
$\overline{X}''=\bigcup_{u\in \overline{\mathcal U}''} u$ and each
$u\in \overline{\mathcal U}''$ has an anchor in $\overline{X}''$.
Finally, no set in $\overline{\mathcal U}'$ contains an anchor
element in $\overline{\mathcal U}''$ since (i) $\overline{\mathcal U}'\subseteq {\mathcal U}'$,
so no set in $\overline{\mathcal U}'$ contains an old anchor element
(one that existed in $X''$), and (ii) no set
in $\overline{\mathcal U}'$
contains one of the new anchors $x$, since each such $x$
was a tail of a nonloop.
The preceding step accomplishes something useful only
if the associated graph is not discrete. If it is,
then we need
A move of type 2.
Suppose now we are at a
stage $(X',X'')$ and $({\mathcal U}', {\mathcal U}'')$
where the associated directed graph is discrete.
If we are not yet to the stage where $X'=\emptyset$,
then choose the element $x\in X'$ that
has least index in our earlier enumeration
$(x_{\lambda})_{\lambda<\kappa}$, and add it
to $X''$. I would like it to be an anchor element,
so choose any $u\in {\mathcal U}'$ that contains $x$
and add $u$ to ${\mathcal U}''$. Also, add
all elements of $u$ to $X''$.
To ensure that $x$
remains
an anchor element
for the rest of the construction, delete all $v\in {\mathcal U}'$
containing $x$. Now the move is complete.
I claim that after these steps
we again have a stage
$(\overline{X}',\overline{X}'')$ and
$(\overline{\mathcal U}', \overline{\mathcal U}'')$,
where the assumptions are satisfied.
To see that $\overline{X}'\subseteq
\bigcup_{u\in \overline{\mathcal U}'} u$
and also that $\overline{\mathcal U}'$
contains no anchor elements in $X''$, 
it suffices
to observe that,
since at the start of the step the associated graph
was discrete, if $y\in \overline{X}'\subseteq X'$, then there
is a $v\in {\mathcal U}'$ such that $y\in v$ and $x\notin v$.
The set $v$ contains none of the old
anchor elements, by induction, and $v$ does not contain
the new anchor element $x$, so $v\in\overline{\mathcal U}'$
and $v$ contains no anchors from $\overline{X}''$.
To see that $\overline{\mathcal U}''={\mathcal U}''\cup\{u\}$
is a minimal
cover of $\overline{X}''=X''\cup u$
it is enough to note that every set in
$\overline{\mathcal U}''$ has an anchor in
$\overline{X}''$.
If we alternate between moves of these two types we will
eventually exhaust $X$, because moves of type 1 cannot 
occur twice in a row and moves of type 2 progress through 
the enumeration of $X$.
We will end with $(\emptyset,X)$ and $(\emptyset, {\mathcal U}_0)$
where ${\mathcal U}_0$ is a minimal cover of $X$.
Last paragraph
As Péter Komjáth pointed out, the inclusion $X'\subseteq \bigcup {\mathcal U}'$ might not be preserved at nonzero limit stages for general $X$, $\mathcal U$. This is not a problem if $|X|=\omega$, since there are no nonzero limit stages. I claim that it is also not a problem if $|X|=\kappa>\omega$ and $\mathcal U$ is a locally finite cover of $X$. Without offering the full details, the reason is that: (i) the inclusion $X'\subseteq \bigcup {\mathcal U}'$ is preserved at successor stages, and (ii) when $\mathcal U$ is a locally finite covering, a point of $X'$ can only be uncovered at a successor stage if our operations only allow moving sets from ${\mathcal U}'$ to ${\mathcal U}''$ or deleting sets from ${\mathcal U}'$.
