Given vectors $V$ of length $d$, construct a graph $G = (V, E)$ where $\{u, v\} \in E$ iff the Pearson correlation between $u$ and $v$ is larger than some threshold $t > 0$. Is $G$ chordal? It seems like it should be, because a long chordless cycle like $a$ correlates with $b$, $b$ with $c$, $c$ with $d$ but nothing else seems difficult to construct. However, I cannot find a simple proof or a reference.



Is the graph of a thresholded correlation matrix chordal?
It was described in [this](https://mathoverflow.net/questions/19406/constructing-bernoulli-random-variables-with-prescribed-correlation) previous question how to obtain a correlation matrix whose entries come from the scalar product of certain vectors $u\_1, u\_2, \dots,u\_n$. If we let the vectors be $$u\_i=(1, \cos(\frac{2\pi i}{n}), \sin(\frac{2\pi i}{n}),0,\dots, 0)$$ we can set a high enough threshold so that the corresponding graph is a cycle of length $n$ and thus not chordal.


