Language of lists of words, not all of which are different, is not context-free

Question

How do I prove that the following language isn't context-free using the pumping lemma? $$ L=\{w_1\#w_2\#\dots\#w_k \colon k ≥ 2, w_i \in \{0,1\}^*, w_i = w_j \text{ for some } i \ne j\} $$

I am having trouble choosing the string to use for the proof. I know that I have to choose a string such that at least two substrings separated by the # are equal to each other but am unsure of how to approach this. If someone could please help me with this, I would appreciate it.

Answer 1

If $L$ were context-free then so would $L' = d(L \cap (0+1)^*\#(0+1)^*)$ be, where $d$ is the homomorphism that deletes $\#$. However, $L'$ is the language of squares (words of the form $w^2$), which is well-known not to be context-free.

If for some reason you have to prove that $L$ is not context-free directly using the pumping lemma, this suggests looking at the proof that $L'$ is not context-free and trying to adapt it.