Article Contents
Article Contents

# Properties of sets of subspaces with constant intersection dimension

• * Corresponding author: Lisa Hernandez Lucas
• A $(k,k-t)$-SCID (set of Subspaces with Constant Intersection Dimension) is a set of $k$-dimensional vector spaces that have pairwise intersections of dimension $k-t$. Let $\mathcal{C} = \{\pi_1,\ldots,\pi_n\}$ be a $(k,k-t)$-SCID. Define $S: = \langle \pi_1, \ldots, \pi_n \rangle$ and $I: = \langle \pi_i \cap \pi_j \mid 1 \leq i < j \leq n \rangle$. We establish several upper bounds for $\dim S + \dim I$ in different situations. We give a spectrum result under certain conditions for $n$, giving examples of $(k,k-t)$-SCIDs reaching a large interval of values for $\dim S + \dim I$.

Mathematics Subject Classification: Primary: 51E20; Secondary: 05B25, 51E23, 94B60.

 Citation:

• Table 1.  Summary of the best bounds found for $\dim S +\dim I$, for different values of $n$, $k$ and $t$

 Condition Upper bound $\dim S + \dim I$ Sharp? Theorem $(k-t)(n-1) \leq k$ $nk$ yes Theorem 2.1 & 2.2 $k\geq 2t$, $n\geq 3$,$(k,n)\neq(2t,3)$ $2k+2(n-2)t-(n-3)$ unknown Theorem 2.5 $k <2t$ $(k-t)(n-1) > k$ $nk$ no Theorem 2.1 & 2.2
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