An Erdős-Ko-Rado theorem for finite classical polar spaces

Abstract

Consider a finite classical polar space of rank d ≥ 2 and an integer n with 0 < n < d. In this paper, it is proved that the set consisting of all subspaces of rank n that contain a given point is a largest Erdős-Ko-Rado set of subspaces of rank n of the polar space. We also show that there are no other Erdős-Ko-Rado sets of subspaces of rank n of the same size.

An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildings

“…For other types of flags in polar spaces, there are very few results with the notable exceptions of [IMM18] which studies flags of type {n−1} in a polar space of rank n, n even, and parameter e = 0 and [Met19a] which investigates flags of type {2} in all polar spaces. We reiterate that the results in [Met16] for k-spaces in polar space of rank n, 1 ≤ k ≤ n, are independent of ours, as the 'far away' relation in that paper is defined as having empty intersection. This is not the same as being opposite.…”

A note on Erdős-Ko-Rado sets of generators in Hermitian polar spaces

“…It turns out that not all these numbers are integers, which is the desired contradiction. The same argument was used in the last section of [6].…”

Cross $t$-intersecting families for symplectic polar spaces

“…A subfamily of P m is called t-intersecting if any two members have a intersection with dimension at least t. The maximum sized t-intersecting subfamilies of P m were widely studied and described. See [14,15] for t = 1 and [13] for all t. Recently, the authors characterized the second largest t-intersecting families [20]. There are also some results for other classical polar spaces, see [2,5,6,12,14,15] for more details.…”

“…See [14,15] for t = 1 and [13] for all t. Recently, the authors characterized the second largest t-intersecting families [20]. There are also some results for other classical polar spaces, see [2,5,6,12,14,15] for more details.…”