Abstract
A cross-intersecting Erdős-Ko-Rado set of generators of a finite classical polar space is a pair $(Y, Z)$ of sets of generators such that all $y \in Y$ and $z \in Z$ intersect in at least a point. We provide upper bounds on $|Y| \cdot |Z|$ and classify the cross-intersecting Erdős-Ko-Rado sets of maximum size with respect to $|Y| \cdot |Z|$ for all polar spaces except some Hermitian polar spaces.
Highlights
Erdos-Ko-Rado sets (EKR sets) were introduced by Erdos, Ko, and Rado [6] as a family of k-element subsets of {1, . . . , n} such that the elements of the family pairwise intersect nontrivially
We provide upper bounds on |Y | · |Z| and classify the crossintersecting Erdos-Ko-Rado sets of maximum size with respect to |Y | · |Z| for all polar spaces except some Hermitian polar spaces
We need some basic properties of an association scheme of generators on a dual polar space of rank d and type e
Summary
Erdos-Ko-Rado sets (EKR sets) were introduced by Erdos, Ko, and Rado [6] as a family of k-element subsets of {1, . . . , n} such that the elements of the family pairwise intersect nontrivially. In polar spaces the problem is partially open, since the maximum size of EKR sets of generators of H(2d − 1, q2), d > 3 odd, is still unknown. There exists the following modification of the original problem which generated a lot of interest: a cross-intersecting EKR set is a pair (Y, Z) of sets of subsets with k elements of {1, . Results for vectors spaces are due to Tokushige [21] This problem can be generalized to polar spaces, where a cross-intersecting EKR set of generators is a pair (Y, Z) of sets of generators such that all y ∈ Y and z ∈ Z intersect in at least a point.
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