Holey graphs: very large Betti numbers are testable

We show that the graph property of having a (very) large $k$-th Betti number $β_k$ 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 $δ(\varepsilon,k)>0$ such that testing whether $β_k \geq (1-δ) d_k$ for $δ\leq δ(\varepsilon,k)$ 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.