Theorems of Erdős–Ko–Rado type in polar spaces

Abstract

We consider Erdős-Ko-Rado sets of generators in classical finite polar spaces. These are sets of generators that all intersect nontrivially. We characterize the Erdős-Ko-Rado sets of generators of maximum size in all polar spaces, except for H(4n + 1, q 2 ) with n 2.

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

“…[3,4,11,13,14]. Given a polar space of rank d and an integer n with 1 ≤ n ≤ d, an Erdős-Ko-Rado set of subspaces of rank n of the polar space is a set of subspaces of rank n of the polar space such that any two subspaces of the set have a non-trivial intersection.…”

“…Pepe et al [13] in their very nice paper have classified the largest Erdős-Ko-Rado sets of generators in all finite classical polar spaces except for H (2d − 1, q) when d ≥ 5 is odd. It should be noted that in some of the polar spaces where the point-pencils are largest Erdős-Ko-Rado sets, there are other Erdős-Ko-Rado sets of the same size.…”

“…One is Q + (2d −1, q) when d ≥ 3 is odd, as already mentioned above. But also in H (5, q), the set consisting of one plane G and all planes meeting G in a line is an Erdős-Ko-Rado set of generators and has more elements than the point-pencils; it was proved in [13] that every largest Erdős-Ko-Rado set of generators of H (5, q) has this form. For all details, we refer to the table at the end of the paper [13].…”

“…It is called a K -regular extended weight matrix, if it has in addition constant row sum K . Hoffman's bound appeared in various forms in the literature, for example in Corollary 3.3 in [6], Theorem 8 in [13], Lemma 6.1 of [8] when applied to the matrix A − θ I , and it was also mentioned in [12]. A proof of the following theorem is well known and is almost identical to the proof of the usual Hoffman's bound.…”

“…Similarly, the induction step for (13) is performed using (9) and the induction hypothesis. We do the induction step for (14) in more detail, where we first use (8) and then the induction hypothesis and also equation (13) for the same value of r − s…”