### On the general position problem on Kneser graphs

#### Abstract

In a graph *G*, a between two vertices *x* and *y* is a shortest path connecting *x* to *y*. A subset *S* of the vertices of *G* is 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 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 $gp(Kn_{n,k})=\binom{n-1}{k-1}$, 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 Bollob'as’s inequality on intersecting set pair systems.

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MANUSCRIPTISSN: 1855-3974

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