The phrase "not give/budge an inch" refers to a person who is stubborn and unwilling to compromise or change their position. Some synonyms for this phrase include "unyielding," "inflexible," "intransigent," and "uncompromising." Another phrase that means the same thing is "digging in one's heels." Such a person is unwilling to move or make concessions, even when it may be in their best interest to do so. These synonyms all convey the idea of someone who is set in their ways and determined to stick to their guns, no matter what.