Property testers are fast randomized algorithms whose task is to distinguish between inputs satisfying some predetermined property and those that are far from satisfying it. A landmark result of Alon et al. states that for any finite family of graphs , the property of being induced ‐free (i.e., not containing an induced copy of any ) is testable. Goldreich and Shinkar conjectured that one can extend this by showing that for any linear inequality involving the densities of the graphs in the input graph, the property of satisfying this inequality is testable. Our main result in this paper disproves this conjecture. The proof deviates significantly from prior nontestability results in this area. The main idea is to use a linear inequality relating induced subgraph densities in order to encode the property of being a quasirandom graph.

中文翻译：

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测试子图统计量的线性不等式

属性测试器是快速随机算法，其任务是区分满足某些预定属性的输入和不满足预定属性的输入。Alon等人的里程碑式的成果。指出对于任何有限的图族，无诱导的属性（即，不包含任何的诱导副本）是可测试的。Goldreich和Shinkar猜想可以通过证明对于涉及图的密度的任何线性不等式可以扩展这一点在输入图中，满足该不等式的性质是可测试的。本文的主要结果证明了这一猜想。该证明与该领域以前的不可测性结果有很大的出入。主要思想是使用与诱导子图密度相关的线性不等式，以编码为拟随机图的性质。