198 research outputs found
The Cameron-Liebler problem for sets
Cameron-Liebler line classes and Cameron-Liebler k-classes in PG(2k+1,q) are
currently receiving a lot of attention. Links with the Erd\H{o}s-Ko-Rado
results in finite projective spaces occurred. We introduce here in this article
the similar problem on Cameron-Liebler classes of sets, and solve this problem
completely, by making links to the classical Erd\H{o}s-Ko-Rado result on sets.
We also present a characterisation theorem for the Cameron-Liebler classes of
sets
Cameron-Liebler sets of k-spaces in PG(n,q)
Cameron-Liebler sets of k-spaces were introduced recently by Y. Filmus and F.
Ihringer. We list several equivalent definitions for these Cameron-Liebler
sets, by making a generalization of known results about Cameron-Liebler line
sets in PG(n, q) and Cameron-Liebler sets of k-spaces in PG(2k + 1, q). We also
present a classification result
Cameron-Liebler k-sets in subspaces and non-existence conditions
In this article we generalize the concepts that were used in the PhD thesis
of Drudge to classify Cameron-Liebler line classes in PG, to
Cameron-Liebler sets of -spaces in PG and AG. In his PhD
thesis, Drudge proved that every Cameron-Liebler line class in PG
intersects every -dimensional subspace in a Cameron-Liebler line class in
that subspace. We are using the generalization of this result for sets of
-spaces in PG and AG. Together with a basic counting argument
this gives a very strong non-existence condition, . This condition
can also be improved for -sets in AG, with
On two non-existence results for Cameron-Liebler -sets in
This paper focuses on non-existence results for Cameron-Liebler -sets. A
Cameron-Liebler -set is a collection of -spaces in or
admitting a certain parameter , which is dependent on the
size of this collection. One of the main research questions remains the
(non-)existence of Cameron-Liebler -sets with parameter . This paper
improves two non-existence results. First we show that the parameter of a
non-trivial Cameron-Liebler -set in should be larger than
, which is an improvement of an earlier known lower
bound. Secondly, we prove a modular equality on the parameter of
Cameron-Liebler -sets in with
, , and even. In
the affine case we show a similar result for and even. This
is a generalization of earlier known modular equalities in the projective and
affine case
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