[PDF][PDF] A Hilton-Milner theorem for vector spaces.
The Electronic Journal of Combinatorics [electronic only], 2010•emis.de
We show for k⩾ 2 that if q⩾ 3 and n⩾ 2k+ 1, or q= 2 and n⩾ 2k+ 2, then any intersecting
family F of k-subspaces of an n-dimensional vector space over GF (q) with⋂ F∈ FF= 0 has
size at most [ n− 1 k− 1]− qk (k− 1)[n− k− 1 k− 1]+ qk. This bound is sharp as is shown by
Hilton-Milner type families. As an application of this result, we determine the chromatic
number of the corresponding q-Kneser graphs.
family F of k-subspaces of an n-dimensional vector space over GF (q) with⋂ F∈ FF= 0 has
size at most [ n− 1 k− 1]− qk (k− 1)[n− k− 1 k− 1]+ qk. This bound is sharp as is shown by
Hilton-Milner type families. As an application of this result, we determine the chromatic
number of the corresponding q-Kneser graphs.
Abstract
We show for k⩾ 2 that if q⩾ 3 and n⩾ 2k+ 1, or q= 2 and n⩾ 2k+ 2, then any intersecting family F of k-subspaces of an n-dimensional vector space over GF (q) with⋂ F∈ FF= 0 has size at most [ n− 1 k− 1]− qk (k− 1)[n− k− 1 k− 1]+ qk. This bound is sharp as is shown by Hilton-Milner type families. As an application of this result, we determine the chromatic number of the corresponding q-Kneser graphs.
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