The Stacks project

Lemma 55.5.16. Nonexistence of proper subgraphs of the form

\[ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \ar@{-}[r] & \bullet \\ & & & & & \bullet } \]

Assume $n > 9$. There do not exist $9$ distinct $(-2)$-indices $d, e, f, g, h, i, j, k, l$ such that $a_{de}, a_{ef}, a_{fg}, a_{gh}, a_{hi}, a_{ij}, a_{jk}, a_{lh}$ are nonzero.

Proof. See discussion above. $\square$

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