Modular and Fractional $L$-Intersecting Families of Vector Spaces

Abstract

This paper is divided into two logical parts. In the first part of this paper, we prove the following 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. 

$\bullet$ Let $V$ be a vector space of dimension $n$ over a finite field of size $q$. Let $K = \{k_1, \ldots , k_r\},L = \{\mu_1, \ldots , \mu_s\}$ be two disjoint subsets of $\{0,1, \ldots , b-1\}$ with $k_1 < \cdots < k_r$.   Let $\mathcal{F} = \{V_1,V_2,\ldots,V_m\}$ be a family of subspaces of $V$ such that (a) for every $i \in [m]$, dim($V_i$) $\bmod~ b = k_t$, for some $k_t \in K$, and (b) for every distinct $i,j \in [m]$, dim($V_i \cap V_j$)$\bmod~ b =  \mu_t$, for some $\mu_t \in L$. Moreover, it is given that neither of the following two conditions hold:  

1. $q+1$ is a power of 2, and $b=2$
2. $q=2, b=6$.

 Then,

$|\mathcal{F}| \leqslant \begin{cases}N(n,s,r,q), & \textrm{ if }\left(s+k_r \leqslant n \textrm { and } r(s-r+1) \leqslant b-1\right) \textrm{ or } (s < k_1 + r)\\ N(n,s,r,q) + \sum_{t \in [r]}\left[\begin{matrix} n \\ k \end{matrix} \right]_{q}, & \textrm{otherwise,} \end{cases}$

where $N(n,s,r,q) := \left[\begin{matrix} n \\ s \end{matrix} \right]_{q} + \left[\begin{matrix} n \\ s-1 \end{matrix} \right]_{q} + \cdots + \left[\begin{matrix} n \\ s-r+1 \end{matrix} \right]_{q}.$  

In the second part of this paper, we prove $q$-analogues of results on a recent notion called fractional $L$-intersecting family of sets for families  of subspaces of a given vector space over a finite field of size $q$. 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.