$$ L=\{a^ib^jc^k \;| \;i, j, k \in \mathbb{N} \; \text{and} \; i <k<j\} $$
I need to show that this language is not context-free with the help of the Pumping Lemma. My first intuition is, that there exist 5 different cases, i.e. the middle part, let's call it vwx, consists of
- only $a$'s
- only $b$'s
- only $c$'s
- $a$'s and $b$'s
- $b$'s and $c$'s
and I need to find a pumping constant, which excludes the new word from the above defined language. However, I am having a hard time how show that formally and precisely. Any hints are highly appreciated!