Nataly Brukhim of the Institute for Advanced Study presents a recent breakthrough in multiclass PAC learning theory, explaining how the optimal sample complexity for learning a hypothesis class reduces to a structural question about Hamming hypergraphs, where vertices lie in [k]^n and edges connect points agreeing on all but one coordinate. She covers the long-standing conjecture that the average degree of these hypergraphs is controlled by the Daniely-Shalev-Shwartz dimension, a 2014 generalization of VC dimension, and how recent work by Chirag Pabbaraju closes a polynomial gap between upper and lower bounds using a linear-algebraic argument. She situates the result alongside her own 2022 work with Carmon, Dinur, Moran, and Yehudayoff, and related contributions from Hanneke, Meng, Moran, and Shaeiri. The talk is aimed at researchers in learning theory and combinatorics.