Concatenate the two tuples to one tuple of length $2ℓ$ and apply the Erdős–Rado theorem for order types of tuples to the induced coloring. This gives a set $H$ of size $2m$ on which the color depends only on the order type of the concatenation. Split $H$ into its $m$ smallest elements $I$ and its $m$ largest elements $J$. Every entry from $I$ lies below every entry from $J$, so for $a ∈ I^ℓ$ and $b ∈ J^ℓ$ the order type of the concatenation is determined by the order types of $a$ and $b$. The function $U(t)$ is the largest $m ≤ t$ such that every $n ≥ t$ admits such sets of size $m$ for every coloring. It is non-decreasing by definition and unbounded by the Ramsey step.

The Ramsey core is adapted from lax-345067 (finite Ramsey theorems for pairs and tuples).