For a field $k$, let $R:=k[x_1, y_1, x_2, y_2...,x_n, y_n...]$, the polynomial ring over $k$ with infinitely many variables. Now let $I_{(a_1, a_2, a_3,...)}$, $a_i\in \{0, 1\}$ for all $i$, be the family of ideals of $R$ generated by a subset of $\{x_1, y_1, x_2, y_2...,x_n, y_n...\}$ as follow:

If $a_1=0$, then $x_1$ is in the generating set of $I_{(a_1, a_2, a_3,...)}$, and if $a_1=1$, then $y_1$ is in the generating set of $I_{(a_1, a_2, a_3,...)}$.

If $a_2=0$, then $x_2$ is in the generating set of $I_{(a_1, a_2, a_3,...)}$, and if $a_2=1$, then $y_2$ is in the generating set of $I_{(a_1, a_2, a_3,...)}$.

\begin{array}{c} . \\ . \\ . \end{array}

For example $I_{(0, 0, 1, 0, 1 ,1, 0 ...)}=\langle x_1, x_2, y_3, x_4, y_5, y_6, x_7...\rangle$.

Now let $\Gamma$ be an infinite subset of $\{0, 1\}^\mathbb{N}$, how can we show that there exists $\alpha\in \Gamma$ such that $I_\alpha\subseteq\cup_{\alpha\neq\beta\in\Gamma}I_\beta$.