Asymptotically sharp polynomial correction to Betti-number bounds

Determine whether, for every fixed homological degree i and positive integer r, there exists a possibly zero polynomial p_i(q) such that every square-free monomial ideal I generated by q elements satisfies β_i(I^r) ≤ binom(binom(q,r), i+1) + p_i(q), with p_i(q) asymptotically negligible relative to the leading term when i=c binom(q,r) for every 0≤c<1.

Background

For r=3, the paper derives an explicit sharp formula for the Betti numbers of E_q3 and consequently a sharp upper bound for the third powers of all square-free monomial ideals. The leading term is the number of faces of the principal square-free facet, while the remaining contributions are polynomial corrections.

The open question asks whether analogous bounds and asymptotic dominance hold for every fixed power r. Such a result would generalize the paper’s third-power estimates and provide substantially sharper bounds than Taylor-type resolutions.

References

For every i, is there a (possibily zero) polynomial p_i(q) such that for every square-free monomial ideal I generated by q elements: βi(Ir) ≤ \binom{\binom{q}{r}}{i+1} + p_i(q) and for every 0 ≤ c < 1, if i = c\binom{q}{r} then \displaystyle \lim{q \rightarrow \infty} \frac{p_i(q)}{\binom{\binom{q}{r}}{i+1}} = 0?

— Realizing resolutions of powers of extremal ideals  (2502.09585 - Chau et al., 13 Feb 2025) in Question 1, part (2), Section 1, Introduction