  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.

Data Structures and Algorithms;Discrete Mathematics;Combinatorics