When Do Stable Matchings Exist?

I just attended MATCHUP 2026 in Paris. There, I learned of the paper “Matching with Attributes” by Aytek Erdil and Battal Dogan (also accepted to EC 2026). This paper gives a creative and satisfying resolution to a longstanding research question: in many-to-one matching with contracts (for concreteness of language, imagine matching doctors to hospitals), what are minimal conditions on hospital…