On the general position problem on Kneser graphs

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 g p ( G ) . Recently, Ghorbani et al. proved that for any k if n ≥ k 3 − k 2 + 2 k − 2 , then g p ( K n n , k ) = ( n − 1 choose k − 1) , where K n n , k denotes the Kneser graph. We improve on their result and show that the same conclusion holds for n ≥ 2.5 k − 0.5 and this bound is best possible. Our main tools are a result on cross-intersecting families and a slight generalization of Bollobas’s inequality on intersecting set pair systems.