Polynomial positive-inertia bound

Determine whether there exists a polynomial f such that every graph G satisfies n^+(G) ≤ f(n^-(G)).

Background

The paper proves an exponential upper bound on n+(G) in terms of n-(G) using Ramsey numbers. The main conjecture would yield a quadratic bound, while the authors note that even establishing an arbitrary polynomial bound would constitute progress toward the broader inertia problem.

References

Does there exist a polynomial $f(x)$ such that for every graph $G$, n+(G)\le f(n-(G)).

— New conjectures on the inertia of graphs  (2508.01163 - Akbari et al., 2 Aug 2025) in Problem 2.3, Section 6

Even the following weaker problem is of interest.

— New conjectures on the inertia of graphs  (2508.01163 - Akbari et al., 2 Aug 2025) in Problem 5.2, Section 5