More on the power of chain rules in context-free grammars

For all $\epsilon > 0$ we prove for every sufficiently large n: There exists a context-free language CL_n with the following properties:

1.

CL_n has a cfg of size O(n).

2.

Each chain rule free cfg for CL_n has size $\Omega(\epsilon^3 n^{3/2-\epsilon})$.