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Probabilistic Extensions of the Erd\H os-Ko-Rado Property
The classical Erd\H os-Ko-Rado (EKR) Theorem states that if we choose a
family of subsets, each of size (k), from a fixed set of size (n (n > 2k)),
then the largest possible pairwise intersecting family has size (t ={n-1\choose
k-1}). We consider the probability that a randomly selected family of size
(t=t_n) has the EKR property (pairwise nonempty intersection) as and
tend to infinity, the latter at a specific rate. As gets large, the
EKR property is less likely to occur, while as gets smaller, the EKR
property is satisfied with high probability. We derive the threshold value for
using Janson's inequality. Using the Stein-Chen method we show that the
distribution of , defined as the number of disjoint pairs of subsets in
our family, can be approximated by a Poisson distribution. We extend our
results to yield similar conclusions for , the number of pairs of subsets
that overlap in exactly elements. Finally, we show that the joint
distribution can be approximated by a multidimensional
Poisson vector with independent components.Comment: 18 page
The Erdős-Ko-Rado properties of various graphs containing singletons
Let G=(V,E) be a graph. For r≥1, let be the family of independent vertex r-sets of G. For vV(G), let denote the star . G is said to be r-EKR if there exists vV(G) such that for any non-star family of pair-wise intersecting sets in . If the inequality is strict, then G is strictly r-EKR.
Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that, if GΓ and 2r is no larger than the number of connected components of G, then G is r-EKR. However, Holroyd and Talbot conjectured that, if G is any graph and 2r is no larger than μ(G), the size of a smallest maximal independent vertex set of G, then G is r-EKR, and strictly so if 2r<μ(G). We show that in fact, if GΓ and 2r is no larger than the independence number of G, then G is r-EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ′Γ. We also confirm the conjecture for graphs in an even larger set Γ″Γ′
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