Finiteness and asymptotic growth of maximal order dimension

Determine whether the supremum of the order dimensions of posets of regions over all finite central simplicial hyperplane arrangements of rank n and all choices of base region is finite, and, if it is finite, determine its asymptotic growth as n tends to infinity.

Background

The paper defines no universal upper bound for the order dimension of simplicial region posets. It establishes only that the dimension can exceed rank, with lower bounds of 5 for type H4 and 7 for type E6; computational results additionally report dimensions for several exceptional Coxeter types.

The proposed function f(n) takes the supremum of the order dimension over all finite central simplicial arrangements of rank n and all choices of base region. The unresolved issue is whether this supremum is finite for each rank and, if so, how rapidly it grows with n.

References

Question 2. Let $f(n)$ be the supremum of $\dim P(\mathcal A,B)$ over finite central simplicial arrangements of rank $n$ and all choices of $B$. Is $f(n)$ finite, and, if so, what is its asymptotic growth?

— Order dimension beyond rank for simplicial hyperplane arrangements  (2608.14092 - Poliakova, 14 Aug 2026) in Introduction, paragraph beginning “Our results raise two natural questions” (Question 2)