A Common Generalization to Theorems on Set Systems with $\mathcal{L}$-intersections

TL;DR

This paper unifies and generalizes several classical theorems on set systems with intersection properties, providing new bounds and results applicable to both set families and subspace families over finite fields.

Contribution

It introduces a unified framework that generalizes multiple intersection theorems, leading to stronger bounds and extending results to vector space subfamilies over finite fields.

Findings

Unified generalization of classical intersection theorems

Stronger bounds for set systems with $ ext{L}$-intersections

Extension of results to subspace families over finite fields

Abstract

In this paper, we provide a common generalization to the well-known Erd\H{o}s-Ko-Rado Theorem, Frankl-Wilson Theorem, Alon-Babai-Suzuki Theorem, and Snevily Theorem on set systems with $\mathcal{L}$-intersections. As a consequence, we derive a result which strengthens substantially the well-known theorem on set systems with $k$-wise $\mathcal{L}$-intersections by F$\ddot{u}$redi and Sudakov [J. Combin. Theory, Ser. A (2004) 105: 143-159]. We will also derive similar results on $\mathcal{L}$-intersecting families of subspaces of an $n$-dimensional vector space over a finite field $\mathbb{F}_{q}$, where $q$ is a prime power.