Erdős Problem #46 — Monochromatic unit-fraction representations of 1

An old problem of R.L. Graham and myself states: Is it true that if $m_k$ is sufficiently large and we colour the integers $2 \leq t \leq m_k$ by $k$ colours then

$$1 = \sum \frac{1}{t_i}$$

is always solvable monochromatically? I would like to see a proof that $m_2$ exists. (Clearly $m_k \geq n_k$.)