Alon and Shapira proved that every monotone class (closed under taking subgraphs) of undirected graphs is strongly testable, that is, under the promise that a given graph is either in the class or $\varepsilon$-far from it, there is a test using a constant number of samples (depending on $\varepsilon$ only) that rejects every graph not in the class with probability at least one half, and always accepts a graph in the class. However, their bound on the number of samples is quite large, since they heavily rely on Szemer\'edi's regularity lemma. We study the case of posets and show that every monotone class of posets is easily testable, that is, a polynomial number of samples is sufficient. We achieve this via proving a polynomial removal lemma for posets. We give a simple classification: for every monotone class of posets there is an $h$ such that the class is indistinguishable (every large enough poset in one class is $\varepsilon$-close to a poset in the other class) from the class of posets free of the chain $C_h$. This allows to test every monotone class of posets using $O(\varepsilon^{-1})$ samples. The test has two-sided error, but it is almost complete: the probability to refute a poset in the class is polynomially small in the size of the poset. The analogous results hold for comparability graphs, too.

翻译：Alon和Shapira证明了无向图的每个单调类（在子图意义下封闭）是强可测试的，即：在给定图要么属于该类、要么与它ε-远离的承诺下，存在一个使用常数个样本（仅依赖于ε）的测试，能以至少一半的概率拒绝任何不属于该类的图，并始终接受属于该类的图。然而，由于他们严重依赖Szemerédi正则引理，其样本数量上界相当大。我们研究偏序集的情形，证明每个偏序集单调类是易测试的，即多项式数量的样本就足够了。我们通过证明偏序集的多项式移除引理来实现这一点。我们给出一个简洁的分类：对于每个偏序集单调类，存在一个h使得该类与禁止链C_h的偏序集类不可区分（每个足够大的属于一个类的偏序集都与另一个类中的某个偏序集ε-接近）。这使得我们能够使用O(ε^{-1})个样本测试每个偏序集单调类。该测试具有双侧误差，但几乎是完备的：拒绝类中偏序集的概率随着偏序集规模的增大而以多项式量级减小。类似结果对可比图同样成立。