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

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 \(\mathrm{PG}(n,q)\) or \(\mathrm{AG}(n,q)\) 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 \(\mathrm{PG}(n,q)\) 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 \(\mathrm{PG}(n,q)\) with \(x<\frac{q^{n-k}-1}{q^{k+1}-1}\), \(n\geq 2k+1\), \(n-k+1\geq 7\) and \(n-k\) even. In the affine case we show a similar result for \(n-k+1\geq 3\) and \(n-k\) even. This is a generalization of earlier known modular equalities in the projective and affine case.