Hypergraph regularity and random sampling

TL;DR

This paper proves that large random samples of a regular hypergraph preserve its regularity properties with high probability, enabling efficient property testing in hypergraph combinatorics.

Contribution

It extends the hypergraph regularity and sampling results to hypergraphs, showing sampling preserves regularity with minimal error correction.

Findings

Random samples retain hypergraph regularity with high probability

Error correction in quasirandomness measures can be arbitrarily small

Applications to combinatorial property testing are demonstrated

Abstract

Suppose a $k$-uniform hypergraph $H$ that satisfies a certain regularity instance (that is, there is a partition of $H$ given by the hypergraph regularity lemma into a bounded number of quasirandom subhypergraphs of prescribed densities). We prove that with high probability a large enough uniform random sample of the vertex set of $H$ also admits the same regularity instance. Here the crucial feature is that the error term measuring the quasirandomness of the subhypergraphs requires only an arbitrarily small additive correction. This has applications to combinatorial property testing. The graph case of the sampling result was proved by Alon, Fischer, Newman and Shapira.