General position number of intermediate Kneser graphs

Determine \(\gp(K(n,k))\) for integers \(n,k\) satisfying \(2k+1\leq n<2.5k-0.5\).

Background

For fixed kk, the survey records the exact value $\gp(K(n,k))=\binom{n-1}{k-1}$ when n≥2.5k−0.5n\geq2.5k-0.5, while the value is strictly smaller in the range 2k+1≤n<2.5k−0.52k+1\leq n<2.5k-0.5. The exact value in this intermediate range remains unresolved.

References

As noted in, it remains an open problem to determine $\gp(K(n, k))$ for $2k + 1 \leq n < 2.5k - 0.5$.

— The General Position Problem: A Survey  (2501.19385 - V. et al., 31 Jan 2025) in Section 2, subsection “Kneser graphs”