(m,n)-paracompact

Definition

m and n denote cardinalities.
A space X is said to be (m, n)-paracompact if every open covering of X of cardinality no more than m has a locally-n open refinement.

Reference

M. K. Singal, Some Generalizations of Paracompactness, Proceedings of the Kanpur topological conference, 1968. Academia Publishing House of the Czechoslovak Academy of Sciences, Praha, 1971. pp. 245-263.