so as a consequence of Hochster’s formula, $\Delta_W$ is connected for any $W\subset [n]$ with $|W|=t+2$. Since $\index(I)=t$, [@EGHP Theorem 2.1] implies that there exists a minimal cycle of length $t+3$ in $\bar{G}$, say $C$. We may assume that $V(C)=\{1, 2, \ldots, t+3\}$ and $E(C)=\{\{i, i+1\} \:\ 1\leq i\leq t+2\}\cup \{\{1, t+3\}\}$.

Assume that $n> t+3$. We will show that under this assumption, there exists $W\subset [n]$ such that either $|W|=t+2$ and $(\bar{G})_W$ is disconnected which implies that $\Delta_W$ is disconnected, or $|W|=t+5$ and $\widetilde{H}_2(\Delta_W; K)\neq 0$ which implies that $\beta_{t+1, t+5}(I)\neq 0$, and so in that case $\projdim(I)>t$. Both cases are not possible, and hence it will follow that $n=t+3$.

For the construction of such $W$ we consider two cases. Let $j\in [n]\setminus [t+3]$.

Suppose first that there exists $1\leq i\leq t+3$ such that $\{j, i\}\notin E(\bar{G})$. Let $W=\{j\}\cup V(C) \setminus \{r, s\}$ where $r$ and $s$ are neighbors of $i$ in $C$. So $|W|=t+2$ and $(\bar{G})_W$ is not connected.

Suppose now that $\{j, i\}\in E(\bar{G})$ for all $1\leq i\leq t+3$. Assume that either $[n]\setminus V(C)=\{j\}$ or for all $j' \in [n]\setminus V(C)$ we have $\{j, j'\}\in E(\bar{G})$. Then $j$ is an isolated vertex of $G$, a contradiction, since 