Linear resolution conjecture for higher-dimensional path complexes

Let $L_n$ be the path graph on vertices $\{x_1,\ldots,x_n\}$, and let $\Delta_k^t(L_n)$ denote the associated complex. The preceding result establishes the claim when $n=2k-1$. Linear resolution conjecture for $\Delta_k^t(L_n)$. The Stanley–Reisner ring $k[\Delta_k^t(L_n)]$ has a linear resolution and is Cohen–Macaulay for all $n\ge 2k-1$. This extends the verified case $n=2k-1$; the statement for all $n\ge 2k-1$ is not resolved in the supplied text.