An Algebraic Hypergraph Regularity Lemma

Abstract

Szemerédi's regularity lemma is a powerful tool in graph theory. It states that for every large enough graph, there exists a partition of the edge set with bounded size such that most induced subgraphs are quasirandom.When the graph is a definable set φ(x, y) in a finite field F q , Tao's algebraic graph regularity lemma ([Tao12]) shows that there is a partition of the graph φ(x, y) such that all induced subgraphs are quasirandom and the error bound on quasirandomness is O(q −1/4 ).In this work we prove an algebraic hypergraph regularity lemma for definable sets in finite fields, thus answering a question of Tao. We also extend the algebraic regularity lemma to definable sets in the difference fields (F alg q , x q ) and we offer a new point of view on the geometric content of the algebraic regularity lemma.

Definable Regularity Lemmas for Nip Hypergraphs

“…In an orthogonal direction (i.e. outside of the finite VC-dimension context), polynomial bounds on the size of the partition were obtained by Tao [Tao15] for algebraic hypergraphs of bounded description complexity in large finite fields of growing characteristic, generalized to fixed characteristic in [PS13] and to hypergraphs in [CL22].…”