Is it problematic to not include a reference letter from home institution? So I have done an one year exchange program in the US and one summer research project, which give me three letters (I have asked the professors before I leaved there). However, my friend warns me it is bad if none of the three letters is from my home institution. Will the admission committee regard this as something like a red flag?

By the way my home institution isn't that famous. In case it is useful, I want to apply for phd program in (pure) math.
Its better to have atleast one letter of recommendation from your home institution as you might have completed the course work from there. And therefore better to have someone who can tell about your performance during studies.