Roberts’ conjecture on the monotonicity and convergence of the ratio O_d/O_{d-1}

Determine whether the sequence (O_d/O_{d-1})_{d≥1}, where O_d denotes the number of finite O-sequences of multiplicity d, is decreasing in d and converges to a limit strictly greater than 1 as d tends to infinity.

Background

Let O_d be the number of finite O-sequences of multiplicity d, i.e., Hilbert functions of Artinian standard graded quotients with total sum d. Understanding the asymptotic behavior of O_d, and in particular the ratio O_d/O_{d-1}, is a central question in the enumerative theory of O-sequences.

The paper proves that (O_d)d is sub-Fibonacci and that, if (O_d/O{d-1})_d converges, then its limit is at most the golden ratio. Computational evidence up to d = 60 suggests the ratio is decreasing, but no proof is provided. The long-standing conjecture attributed to L. Roberts asserts that this ratio is indeed decreasing and converges to a limit greater than 1.