Positive Codegree Andrásfai–Erdős–Sós Theorem for the Generalized Triangle

ABSTRACT The celebrated Andrásfai–Erdős–Sós theorem from 1974 shows that every ‐vertex triangle‐free graph with minimum degree greater than must be bipartite. We establish a positive codegree extension of this result for the ‐uniform generalized triangle : For every , if is an ‐vertex ‐free ‐uniform hypergraph in which each ‐tuple of vertices is contained in either zero edges or more than edges of , then is ‐partite. This result provides the first tight positive codegree Andrásfai–Erdős–Sós type theorem for hypergraphs. It also immediately implies that the positive codegree Turán number of is for all . Additionally, for , our result answers one of the questions posed by Hou et al. in a strong form.