There are various ways to express the concept of "taking up a claim" depending on the context. To assert a right to something, one might say "laying claim to" or "stake a claim." To begin the process of acquiring ownership of a property, one could use phrases like "file an application for a claim" or "initiate a land claim." Alternatively, if the situation involves a dispute with another person, "challenging a claim" could be more appropriate. In essence, the term "taking up a claim" refers to the act of establishing a legitimate right to something, and there are plenty of ways to convey that with different phrasings and contexts.