9 research outputs found
Colouring Lines in Projective Space
Let be a vector space of dimension over a field of order . The
-Kneser graph has the -dimensional subspaces of as its vertices,
where two subspaces and are adjacent if and only if
is the zero subspace. This paper is motivated by the problem
of determining the chromatic numbers of these graphs. This problem is trivial
when (and the graphs are complete) or when (and the graphs are
empty). We establish some basic theory in the general case. Then specializing
to the case , we show that the chromatic number is when and
when . In both cases we characterise the minimal
colourings.Comment: 19 pages; to appear in J. Combinatorial Theory, Series
Shadows and intersections in vector spaces
AbstractWe prove a vector space analog of a version of the Kruskal–Katona theorem due to Lovász. We apply this result to extend Frankl's theorem on r-wise intersecting families to vector spaces. In particular, we obtain a short new proof of the Erdős–Ko–Rado theorem for vector spaces