4 problems

Limit-order conjecture.

Let $K[X^1,\dots,X^k]$ be the polynomial ring in $k$ sets of countably many variables, with $\operatorname{Sym}(\infty)$ permuting the indices. Let $I\subset K[X^1,\dots,X^k]$ be a…

Let $K[X]$ be the polynomial ring in countably many variables, with the symmetric group acting by permuting variables. A homogeneous ideal $I\subset K[X]$ is symmetric if it is inv…

Let $k$ be an infinite field of positive characteristic, and let … be a chain of homogeneous symmetric ideals with $I_n \subset k[x_1,x_2,\ldots,x_n]$. For each $n$, consider the r…