Let $X$ be the first uncountable ordinal. In other words $X$ is an uncountable set equipped with a well-ordering relation "$\leq $" such that, for every $x$ in $X$, the set $$ \{y\in X:\ y\leq x\} $$ is countable. Let $$ T=\{(x, y)\in X\times X:\ y\leq x \}, $$ so that $T$ consist of everything below the diagonal in $X\times X$.

Let us agree to call a subset $G\subseteq T$ a *graph*, provided
$$
\big ((x,y_1)\in G\big ) \ \wedge\ \big ((x,y_2)\in G\big ) \ \Rightarrow \ y_1=y_2.
$$
Clearly these are precisely the graphs of $X$-valued functions $f$ defined on subsets of $X$, such that $f(x)\leq x$, for every $x$.

Question: Is it possible to write $T$ as the union of a countable family of graphs?