Cameron–Liebler k-sets in subspaces and non-existence conditions

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In this article we generalize the concepts that were used in the PhD thesis of Drudge to classify Cameron–Liebler line classes in PG $$(n,q), n\ge 3$$ , to Cameron–Liebler sets of k-spaces in $${{\,\mathrm{\mathrm {PG}}\,}}(n,q)$$ and $${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$$ . In his PhD thesis, Drudge proved that every Cameron–Liebler line class in $${{\,\mathrm{\mathrm {PG}}\,}}(n,q)$$ intersects every 3-dimensional subspace in a Cameron–Liebler line class in that subspace. We are using the generalization of this result for sets of k-spaces in $${{\,\mathrm{\mathrm {PG}}\,}}(n,q)$$ and $${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$$ . Together with a basic counting argument this gives a very strong non-existence condition, $$n\ge 3k+3$$ . This condition can also be improved for k-sets in $${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$$ , with $$n\ge 2k+2$$ .