Unbounded regularity in fixed homological degree

Construct, for each fixed homological degree \(i\ge 2\), graphs \(\Gamma\) whose homology modules \(H_i(B\Gamma)\) have arbitrarily large regularity as \(F[E]\)-modules.

Background

Regularity controls the stable range in which the Betti-number sequence agrees with its Hilbert polynomial. The paper notes that arbitrarily high regularity could have consequences for the existence of finite universal presentations. It leaves unresolved whether regularity can grow without bound when the homological degree is fixed.

References

For a fixed i\ge 2, is there a graph \Gamma with arbitrarily large regularity of H_i(B\Gamma)?

— Hilbert polynomials of configuration spaces over graphs of circumference at most 1  (2505.24416 - An et al., 30 May 2025) in Section 1, subsection “Future directions,” item 3