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Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990s.
The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set ${1,2,dots, n}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partiti
The abstract induced subgraph poset of a graph is the isomorphism class of the induced subgraph poset of the graph, suitably weighted by subgraph counting numbers. The abstract bond lattice and the abstract edge-subgraph poset are defined similarly b
The Stern poset $mathcal{S}$ is a graded infinite poset naturally associated to Sterns triangle, which was defined by Stanley analogously to Pascals triangle. Let $P_n$ denote the interval of $mathcal{S}$ from the unique element of row $0$ of Sterns
Let $H$ be connected $m$-uniform hypergraph and $mathcal{A}(H)$ be the adjacency tensor of $H$. The stabilizing index of $H$, denoted by $s(H)$, is exactly the number of eigenvectors of $mathcal{A}(H)$ associated with the spectral radius, and the cyc
Let H be a hypergraph on n vertices with the property that no edge contains another. We prove some results for a special case of the Isolation Lemma when the label set for the edges of H can only take two values. Given any set of vertices S and an ed