A quantitative container characterization of one-sided testability

2026-08-02

AI Summary

This work resolves an open problem posed by Alon et al. concerning a quantitative combinatorial characterization of scale-independent one-sided testability in dense graph models. By establishing a quantitative equivalence between one-sided testability and the existence of hypergraph containers, the paper provides the first unified quantitative characterization applicable to hereditary graph properties and, via semi-heredity, extendable to arbitrary graph properties. The approach avoids reliance on Szemerédi’s regularity lemma and applies broadly to finite relational structures such as directed graphs, edge-colored graphs, and hypergraphs, enabling explicit translation between tester complexity and container parameters. As applications, the authors derive quantitative closure properties for partitionable graph properties and construct efficient testers for properties admitting linear-sized induced substructures.

Abstract

We give a quantitative combinatorial characterization of size-oblivious one-sided testability in the dense graph model, resolving a question of Alon, Fischer, Newman, and Shapira. For hereditary graph properties, we prove that one-sided testability is quantitatively equivalent to the existence of suitable hypergraph containers, a central and widely used tool in modern combinatorics. Combining this equivalence with the Alon-Shapira notion of semi-hereditariness yields a quantitative characterization of arbitrary graph properties. The correspondence is effective in both directions and provides explicit translations between tester complexity and container parameters. Our proof is regularity-free and extends uniformly to every fixed finite relational signature of bounded arity, including digraphs, coloured graphs, and hypergraphs. As applications, we obtain quantitative closure results for partition properties and testers for properties defined by the existence of a linearly large induced substructure.

Problem

• one-sided testability
• hereditary graph properties
• hypergraph containers
• dense graph model
• quantitative characterization

Innovation

• one-sided testability
• hypergraph containers
• hereditary graph properties
• regularity-free proof
• quantitative characterization