# On the general position problem on Kneser graphs

## TL;DR

This paper improves the bounds on the size of the largest set of vertices in general position in Kneser graphs, showing the previous bound can be lowered from cubic to linear in k, with optimality.

## Contribution

The authors establish a tighter bound on the general position number of Kneser graphs, extending previous results and proving the new bound is best possible.

## Key findings

- The bound n ≥ 2.5k - 0.5 is sufficient for the known general position number.
- The previous bound n ≥ k^3 - k^2 + 2k - 2 is improved.
- The new bound is proven to be optimal.

## Abstract

In a graph $G$, a geodesic between two vertices $x$ and $y$ is a shortest path connecting $x$ to $y$. A subset $S$ of the vertices of $G$ is in general position if no vertex of $S$ lies on any geodesic between two other vertices of $S$. The size of a largest set of vertices in general position is the general position number that we denote by $gp(G)$. Recently, Ghorbani et al, proved that for any $k$ if $n\ge k^3-k^2+2k-2$, then $gp(Kn_{n,k})=\binom{n-1}{k-1}$, where $Kn_{n,k}$ denotes the Kneser graph. We improve on their result and show that the same conclusion holds for $n\ge 2.5k-0.5$ and this bound is best possible. Our main tools are a result on cross-intersecting families and a slight generalization of Bollob\'as's inequality on intersecting set pair systems.