Search or compare within a Grapheme Cluster in Korean

I'm no expert, but I think you're very unlikely to find a clean solution for what you want. There doesn't seem to be any relationship between a Korean character's Unicode value and the graphemes that it's made up of.

e.g. "이" is \uc774 and "ㅣ" is \u3163. From the perspective of the NSString, they're just two different characters with no specific relationship to each other.

I suspect that you will have to find or create an explicit mapping between characters and their graphemes, and then write your own search function that consults this mapping.

This very long page on Unicode Korean can help you, if it comes to that. It has a table of all the characters which suggests some structured relation between the way characters are numbered and their components.