Fluctuation of associated primes for square-free monomial ideals in six variables

Determine whether there exists a square-free monomial ideal in the polynomial ring K[x1, x2, x3, x4, x5, x6] whose powers exhibit fluctuation in their sets of associated primes, thereby resolving the remaining six-variable case in the classification of fluctuation phenomena for square-free monomial ideals.

Background

The paper establishes that for every n ≥ 3 there are infinitely many general monomial ideals in K[x1, ..., xn] whose powers exhibit fluctuation in their associated-prime sets. For square-free monomial ideals, it records that no fluctuation occurs when n ≤ 5 because such ideals satisfy the strong persistence property, while examples in seven variables can be extended to every n ≥ 7 using the expansion construction.

Consequently, the existence of a fluctuating square-free monomial ideal remains unresolved only in six variables. A positive answer would complete the classification of fluctuation phenomena for square-free monomial ideals across all polynomial rings; a negative answer would establish that six variables, like at most five variables, preclude such fluctuations.