Bounds on the sizes of extremal families on finite sets 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 mathbbFq^nover the finite field mathbbFqwhich gives a q-analogue of the ErdHos k-Sperner 定理, and we then establish a general relationship between upper bounds for the sizes of families of subsets of [n] = 1, 2, …, nwith 性质 P and upper bounds for the sizes of families of subspaces of mathbbFq^nwith 性质 P, where P is either mathcalL-intersecting or forbidding certain configurations. Applying this relationship, we derive generalizations of the well-known results about the famous ErdHos matching 猜想 and ErdHos-Chvátal simplex 猜想 to linear lattices.