5 problems

Let $S=K[x_1,\ldots,x_n]$, let $J_{n,m}$ be the path ideal of the cycle graph on $n$ vertices with paths of length $m$, and set $d=\gcd(n,m)$. Let $t_0\leq n-1$ be maximal such tha…

Let $T$ be a tree, let $t$ be a positive integer, and let $J_t(T)$ denote the $t$-connected ideal of $T$. Let $\operatorname{bight}(J_t(T))$ denote the big height of this ideal. Pr…

Let $T$ be a tree, let $t$ be a positive integer, and let $J_t(T)$ denote the $t$-connected ideal of $T$. Let $\mathcal H$ be the hypergraph corresponding to $J_t(T)$, and write…

Regularity conjecture for t-path ideals. The displayed formula should hold. In particular,

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring, let $J_{n,m}$ denote the path ideal of the cycle graph on $n$ vertices with path length $m$, and let $\varphi(n,m)$ be the functio…