A question on subsets of $\omega_1$ Is there a family $\{A_\alpha:\alpha<2^{\omega_1}\}\subset [\omega_1]^{\omega_1}$ in ZFC such that for each  countable set $I\subset 2^{\omega_1}$ and $\alpha\in 2^{\omega_1}\setminus I$ we have
$$A_\alpha\not\subset\bigcup\{A_\beta:\beta\in I\}.$$
Of course, we can find such a family of size $\omega_2$ instead of $2^{\omega_1}$.
 A: Today I could prove  that the existence of such family is not provable from  ZFC. 
Theorem.
If GCH holds in $V$,  and $\mu\ge \omega_3$  then in $V^{Fn(\mu,2)}$ the following holds:
If $\{A_i:i<{\omega}_3\}\subset [{\omega}_1]^{{\omega}_1}$, then there is
 $I\in [\omega_3]^\omega$ and $\alpha\in \omega_3\setminus I $ such that 
 $$A_{\alpha}\subset \bigcup_{i\in I}A_i.$$
Proof.
An ${Fn(I,2)}$-name $\dot B$ of a subset of $\omega_1$ is called 
 nice if for each ${\nu}<\omega_1$ there is an antichain 
$B_{\nu}\subset  {Fn(I,2)}$ such that 
$$
\dot B=\{\langle p,\check{{\nu}}\rangle:{\nu}\in {\omega_1}\land p\in B_{\nu}\}=
\bigcup\{B_{\nu}\times\{\check{{\nu}}\}:{\nu}\in {\omega_1}\}.
$$
We let $supp(\dot B)=\bigcup\{dom(p):p\in\bigcup\limits_{{\nu}<{\omega_1}}B_{\nu}\}$.
It is well-known  that every set of ordinals in  $V[G]$ has a nice name in $V$.
If ${\varphi}$ is a bijection between two sets $I$ and $J$ then
${\varphi}$ lifts to a natural isomorphism between ${Fn(I,2)}$ and
${Fn(I,2)}$, which will be also denoted by ${\varphi}$, as follows: 
for $p\in {Fn(I,2)}$ let $dom({\varphi}(p))={\varphi}''dom(p)$ and
${\varphi}(p)({\varphi}({\xi}))=p({\xi})$. Moreover ${\varphi}$ also
generates a bijection between the nice ${Fn(I,2)}$-names and the nice
$Fn(J,2)$-names: 
if $\dot A$ is a nice $Fn(I,2)$-name then let
${\varphi}(\dot A  )=\{\langle{\varphi}(p),\hat{{\xi}}\>:\langle p,\hat{{\xi}}\rangle \in\dot B\}$.
If $I$ and $J$ are sets of ordinals with the same order type then
  ${\varphi}_{I,J}$ is the natural order-preserving
bijection from $I$ onto $J$.
For each $i<{\omega}_3$ pick a nice $Fn(\mu,2)$-name $\dot A_i$ 
for $A_i$ in $V$. Then  $S_i=supp(A_i)\in [\mu]^{\le \omega_1}$.
Since $2^{\omega_1}=\omega_2$ in $V$ there is a subset 
$K\in [\omega_3]^{\omega_3}$ such that 
(1) $\{S_i:i\in K\}$ forms a $\Delta$-system with kernel $S$, 
(2) every element  of the family $\{S_i:i\in K\}$ has the same
    order type  
(3) $\varphi_{S_i,S_j}$ is the identity on $S$ for all
    $i,j\in K$.  
(4) $\varphi_{S_i,S_J}(\dot A_i)=\dot A_j$ for
    $i,j\in K$.
Let $I\in [K]^\omega$ and $\alpha \in K\setminus I$.
Assume on the contrary that  $p\in Fn(\mu,2)$ and $\zeta<\omega_1$ such that 
$$
p\Vdash \check \zeta\in \dot A_{\alpha}\setminus \bigcup_{i\in I}\dot A_i.  
$$
Pick $\beta \in I$ such that $dom(p)\cap (S_\beta\setminus S)=\emptyset.$
Let 
$$
r=p\cup \varphi_{S_\alpha, S_\beta}(p\restriction S_\alpha).
$$
Since $dom(p)\cap (S_\beta\setminus S)=\emptyset$, $r$ is a condition.
Moreover, $p\Vdash \check \zeta\in \dot A_{\alpha}$,
so $p\restriction S_\alpha \Vdash \check \zeta\in \dot A_{\alpha}$,
and so $\varphi_{S_\alpha, S_\beta}(p\restriction S_\alpha )\Vdash 
\check \zeta\in \dot A_{\beta}$. 
Thus $r\Vdash \zeta\in \bigcup_{i\in I}\dot A_i.$  Contradiction.
