On symmetric hollow integer matrices with eigenvalues bounded from below

A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define $\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$, where $\rho$ is the unique real root of $x^3 = x + 1$. We show that for every $\lambda < \lambda^*$, there exists $n \in \mathbb{N}$ such that if a symmetric hollow integer matrix has an eigenvalue less than $-\lambda$, then one of its principal submatrices of order at most $n$ does as well. However, the same conclusion does not hold for any $\lambda \ge \lambda^*$.