EIGENVALUES AND LINEAR QUASIRANDOM HYPERGRAPHS

John Lenz,Dhruv Mubayi

doi:10.1017/fms.2014.22

Abstract

Let$p(k)$denote the partition function of$k$. For each$k\geqslant 2$, we describe a list of$p(k)-1$quasirandom properties that a$k$-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness by Kohayakawa, Rödl, and Skokan, and by Conlon, Hàn, Person, and Schacht, and the spectral approach of Friedman and Wigderson. For each of the quasirandom properties that is described, we define the largest and the second largest eigenvalues. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of Conlon, Hàn, Person, and Schacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of Chung, Graham, and Wilson for graphs.

Highlights

The study of quasirandom or pseudorandom graphs was initiated by Thomason [43, 44] and refined by Chung et al [15], resulting in a list of equivalent properties of graph sequences which are inspired by G(n, p). Beginning with these foundational papers on the subject [15, 43, 44], the last two decades have seen an explosive growth in the study of quasirandom structures in mathematics and computer science
We refer the reader to a survey of Krivelevich and Sudakov [30] for graphs, and recent papers of Gowers [23,24,25] for general quasirandom structures including hypergraphs
We define a generalization of Eig to k-uniform hypergraphs and add it into the equivalences stated in Theorem 1

Summary

Introduction

The study of quasirandom or pseudorandom graphs was initiated by Thomason [43, 44] and refined by Chung et al [15], resulting in a list of equivalent (deterministic) properties of graph sequences which are inspired by G(n, p). We define a generalization of Eig to k-uniform hypergraphs and add it into the equivalences stated in Theorem 1. Count[π -linear]: If F is an f -vertex, m-edge, k-uniform, π -linear hypergraph, the number of labeled copies of F in Hn is pmn f + o(n f ).

Results

Conclusion