Counting results for sparse pseudorandom hypergraphs II

TL;DR

This paper extends universality results to sparse 3-uniform hypergraphs within strongly jumbled hypergraphs, introducing a counting lemma for fixed hypergraphs in pseudorandom hypergraphs.

Contribution

It provides a new universality result for sparse hypergraphs and a counting lemma for fixed hypergraphs in pseudorandom hypergraphs, building on prior work.

Findings

Establishes a universality result for sparse 3-uniform hypergraphs

Proves a counting lemma for fixed hypergraphs in pseudorandom hypergraphs

Advances understanding of hypergraph universality and pseudorandomness

Abstract

We present a variant of a universality result of R\"odl [On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1986), no. 1-2, 125-134] for sparse, $3$-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ``pseudorandom'' uniform hypergraphs, which is proved in the companion paper [Counting results for sparse pseudorandom hypergraphs I].