$\mathcal{L}$-intersecting or Configuration Forbidden Families on Set Systems and Vector Spaces over Finite Fields

Abstract

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 ${\mathbb{F}}_q^n$ over finite field ${\mathbb{F}}_q$ which gives a $q$-analogue of the Erdős' $k$-Sperner Theorem, and we then establish a general relationship between upper bounds for the sizes of families of subsets of $[n]=\{1,2,\dots ,n\}$ with property $P$ and upper bounds for the sizes of families of subspaces of ${\mathbb{F}}_q^n$ with property $P$, where $P$ is either $\mathcal{L}$-intersecting or forbidding certain configuration. Applying this relationship, we derive generalizations of the well known results about the famous Erdős matching conjecture and Erdős-Chvátal simplex conjecture to linear lattices. As a consequence, we disprove a related conjecture on families of subspaces of ${\mathbb{F}}_q^n$ by Ihringer [Europ. J. Combin., 94 (2021), 103306].