7 problems

EHHM's conjecture. The function $g_I(k)$ is nonincreasing for every square-free monomial ideal $I$.

Erey–Herzog–Hibi–Saeedi Madani conjecture. For any squarefree monomial ideal $I$, the function $g_I(k)$ is nonincreasing.

Nonincreasing normalized depth function conjecture. The function $g_I$ is nonincreasing.

Monotonicity conjecture. For any squarefree monomial ideal, $g_I(k)$ is a non-increasing function. The conjecture concerns the behavior of normalized depth under squarefree powers,…

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$, let $I\subset S$ be a suitable homogeneous ideal, and let $f_I(k)=\operatorname{depth}(S/I^k)$. A function…

Normalized-depth conjecture. The function $g_I(k)$ is nonincreasing.

Let $I$ be a monomial ideal in a polynomial ring $S$, and write $f(k)=\operatorname{depth}(S/I^k)$ for its depth function. Depth-function conjecture. The function $f(k)$ can be any…