On two non-existence results for Cameron–Liebler k-sets in $${{\,\mathrm{\textrm{PG}}\,}}(n,q)$$

Abstract

This paper focuses on non-existence results for Cameron–Liebler k-sets. A Cameron–Liebler k-set is a collection of k-spaces in \({{\,\mathrm{\textrm{PG}}\,}}(n,q)\) or \({{\,\mathrm{\textrm{AG}}\,}}(n,q)\) admitting a certain parameter x, which is dependent on the size of this collection. One of the main research questions remains the (non-)existence of Cameron–Liebler k-sets with parameter x. This paper improves two non-existence results. First we show that the parameter of a non-trivial Cameron–Liebler k-set in \({{\,\mathrm{\textrm{PG}}\,}}(n,q)\) should be larger than \(q^{n-\frac{5k}{2}-1}\), which is an improvement of an earlier known lower bound. Secondly, we prove a modular equality on the parameter x of Cameron–Liebler k-sets in \({{\,\mathrm{\textrm{PG}}\,}}(n,q)\) with \(x<\frac{q^{n-k}-1}{q^{k+1}-1}\), \(n\ge 2k+1\), \(n-k+1\ge 7\) and \(n-k\) even. In the affine case we show a similar result for \(n-k+1\ge 3\) and \(n-k\) even. This is a generalization of earlier known modular equalities in the projective and affine case.