L-intersecting or Configuration Forbidden Families on Set Systems and Vector Spaces over Finite Fields

In this paper, we derive a tight upper bound for the size of an intersecting k-Sperner family of subspaces of the n-dimensional vector space Fqⁿ over finite field Fq which gives a q-analogue of the Erd{o}s' k-Sperner Theorem, and we then establish a general relationship between upper bounds for the sizes of families of subsets of [n] = \1, 2, , n\ with property P and upper bounds for the sizes of families of subspaces of Fqⁿ with property P, where P is either L-intersecting or forbidding certain configuration. Applying this relationship, we derive generalizations of the well known results about the famous Erd{o}s matching conjecture and Erd{o}s-Chv\'atal simplex conjecture to linear lattices. As a consequence, we disprove a related conjecture on families of subspaces of Fqⁿ by Ihringer [Europ. J. Combin., 94 (2021), 103306].