We show that the graph property of having a (very) large $k$-th Betti number $\beta_k$ for constant $k$ is testable with a constant number of queries in the dense graph model. More specifically, we consider a clique complex defined by an underlying graph and prove that for any $\varepsilon>0$, there exists $\delta(\varepsilon,k)>0$ such that testing whether $\beta_k \geq (1-\delta) d_k$ for $\delta \leq \delta(\varepsilon,k)$ reduces to tolerantly testing $(k+2)$-clique-freeness, which is known to be testable. This complements a result by Elek (2010) showing that Betti numbers are testable in the bounded-degree model. Our result combines the Euler characteristic, matroid theory and the graph removal lemma.

翻译：我们证明，在稠密图模型中，对于常数$k$，图具有（极）大第$k$个Betti数$\beta_k$的性质可用常数次查询进行测试。具体而言，我们考虑由底层图定义的团复形，并证明对于任意$\varepsilon>0$，存在$\delta(\varepsilon,k)>0$，使得当$\delta \leq \delta(\varepsilon,k)$时，测试$\beta_k \geq (1-\delta) d_k$可归约为容错测试$(k+2)$-团自由性（该性质已知可测试）。此结果补充了Elek（2010）在有限度模型中对Betti数可测试性的证明。我们的证明结合了欧拉示性数、拟阵论和图移除引理。