Modular and fractional L-intersecting families of vector spaces

TL;DR

This paper extends classical intersection theorems to vector spaces over finite fields, introducing modular and fractional L-intersecting families, and provides new bounds on their sizes with general and improved results.

Contribution

It proves a q-analogue of a generalized modular Ray-Chaudhuri-Wilson theorem and establishes bounds for fractional L-intersecting families of subspaces, advancing combinatorial theory.

Findings

Proved a q-analogue of a generalized modular Ray-Chaudhuri-Wilson theorem.

Derived upper bounds for the size of fractional L-intersecting families.

Improved bounds in specific cases for these families.

Abstract

In the first part of this paper, we prove a theorem which is the $q$-analogue of a generalized modular Ray-Chaudhuri-Wilson Theorem shown in [Alon, Babai, Suzuki, J. Combin. Theory Series A, 1991]. It is also a generalization of the main theorem in [Frankl and Graham, European J. Combin. 1985] under certain circumstances. In the second part of this paper, we prove $q$-analogues of results on a recent notion called \emph{fractional $L$-intersecting family} for families of subspaces of a given vector space. We use the above theorem to obtain a general upper bound to the cardinality of such families. We give an improvement to this general upper bound in certain special cases.