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 $.

中文翻译：

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具有恒定交点尺寸的子空间集的性质

$（k，kt）$ -SCID（具有恒定交点维的子空间集）是一组$ k $维向量空间，这些向量空间具有成对相交的维$ kt $。假设$ \ mathcal {C} = \ {\ pi_1，\ ldots，\ pi_n \} $为$（k，kt）$ -SCID。定义$ S：= \ langle \ pi_1，\ ldots，\ pi_n \ rangle $和$ I：= \ langle \ pi_i \ cap \ pi_j \ mid 1 \ leq i <j \ leq n \ rangle $。在不同情况下，我们为$ \ dim S + \ dim I $建立了几个上限。我们给出在特定条件下$ n $的频谱结果，并举例说明$（k，kt）$ -SCID达到$ \ dim S + \ dim I $的较大值区间的示例。