Determine the diagonal threshold for unions of intervals

Determine whether the eventual constant value \(\kappa_s\) of the diagonal Hadwiger–Debrunner numbers \(HD_1^{(s)}(t,t)\) equals \(s\) for every integer \(s\ge 2\).

Background

For a fixed number $s\ge2$ of interval components, the paper proves that the diagonal quantities $H_s(t,t)=HD_1^{(s)}(t,t)$ eventually stabilize at an integer $\kappa_s$ satisfying $s\le \kappa_s\le 2s$. The same parameter controls the two possible asymptotic values of $HD_1^{(s)}(p,q)$ in the logarithmic range: these values are $p-q+\kappa_s$ and $p-q+\kappa_s+1$.

The exact value of $\kappa_s$ is not determined by the argument. Establishing whether the lower endpoint $\kappa_s=s$ holds for every number of components would identify the diagonal obstruction and sharpen the concentration result.