Consider any sequence \( a_1,\dots, a_n \) of non-negative integers in \(\{0,1,\dots, m\}\). Prove that \[|\{ a_i+ a_j + (j-i): 1\leq i < j \leq n \}|\geq m \] when \(m= \lfloor \frac{1}{4} n^{2/3} \rfloor \).

A bonus problem: Can you find a function \(f(n)=\omega(n^{2/3})\) such that the above statement is true when \(m = f(n) \)? Is there such a function with \(f(n)= \Omega(n)\)? (You would still get full points without answering the bonus question.)