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On the connectivity of random subsets of projective spaces. (English) Zbl 0939.51012
Let $$PG(r-1,q)$$ be the projective space of dimension $$r-1$$ over the Galois field $$GF(q)$$. Let $$S$$ be a subset of points of $$PG(r-1,q)$$. The subset $$S$$ is said to be independent if it spans a subspace of dimension $$|S|-1$$. If $$T$$ is any subset of points of $$PG(r-1,q)$$, the rank $$\rho (T)$$ of $$T$$ is the size of the largest independent set contained in $$T$$.
The pair $$(PG(r-1,q), \rho)$$ can be viewed as a matroid and so it is natural to consider the connectivity of subsets of $$PG(r-1,q)$$. A subset $$T$$ of $$PG(r-1,q)$$ such that $$\rho (T) =r$$ is said to be $$k$$-separable for $$k\geq 1$$ if there exists a separator $$S\subseteq T$$ such that $$|S|\leq k$$ and $$\rho (T\setminus S)<r$$. If $$k\geq 2$$ and $$T$$ is not $$k-1$$-separable then $$T$$ is said to be $$k$$-connected.
The main result in the paper under review is that with probability tending to one as $$r$$ tends to infinity, a random subset $${\omega}_ r(n)$$ of cardinality $$n$$ of $$PG(r-1,q)$$ becomes $$k$$-connected when $$n=r+(k-1)\log _q(r) + O(1)$$.

##### MSC:
 51E20 Combinatorial structures in finite projective spaces 05B35 Combinatorial aspects of matroids and geometric lattices 05B25 Combinatorial aspects of finite geometries
##### Keywords:
random subsets; connectivity; matroid
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##### References:
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