Eigenvalues and Quasirandom Hypergraphs

Let p(k) denote the partition function of k. For each k ≥ 2, we describe a list of p(k) − 1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on hypergraph quasirandomness, beginning with the early work of Chung and Graham and Frankl-Rödl related to strong hypergraph regularity, the spectral approach of Friedman-Wigderson, and more recent results of KohayakawaRödl-Skokan and Conlon-Hàn-Person-Schacht on weak hypergraph regularity and its relation to counting linear hypergraphs. For each of the quasirandom properties that are described, we define a hypergraph eigenvalue analogous to the graph case and a hypergraph extension of a graph cycle of even length whose count determines if a hypergraph satisfies the property. This answers a question of Conlon et al. Our work can be viewed as an extension to hypergraphs of the seminal results of Chung-Graham-Wilson for graphs.