The poset of hypergraph quasirandomness

Abstract

Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-GrahamWilson 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 1990's.

Perfect packings in quasirandom hypergraphs I

“…• More generally, when I is a partition the property Disc (k) (I, p, µ) is essentially the property Expand[π] studied in [29,26,28]. In particular, when I = {{1, .…”

“…• When I = [k] , then Disc (k) (I, p, µ) is closely related to the property CliqueDisc[ ] studied in [2,3,4,5,6,19,28].…”

“…As mentioned previously, to show that A (k) n satisfies Disc (k) (I, ≥p, µ), we combine Lemma 12 with a theorem of Towsner [38] which is stated in the language of k-graph sequences. Converting from the probability distribution A (k) n to a k-graph sequence is very similar to the proofs of [28,Lemmas 30 and 31] so we only briefly sketch the technique here. By the previous lemmas and the probabilistic method, for every µ > 0 there exists an n 0 such that for every n ≥ n 0 there exists some k-graph satisfying the properties in the previous lemmas (has the right edge density, fails Disc (k) (…”

“…There is a rather well-defined notion of quasirandomness for graphs that originated in early work of Thomason [36,37] and Chung-Graham-Wilson [7]. These graph quasirandom properties, when generalized to k-graphs, provide a rich structure of inequivalent hypergraph quasirandom properties (see [28,38]). In [27], the authors studied in detail the packing problem for the simplest of these quasirandom properties, the so-called weak hypergraph quasirandomness.…”

A Combinatorial Classic — Sparse Graphs with High Chromatic Number

“…But in passing let us stress that no full analogy of graphs X p,q is known for hypergraphs (see for example recent [54]). In particular, no small explicit construction of hypergraphs is known.…”