The second largest Erdős–Ko–Rado sets of generators of the hyperbolic quadrics Q⁺(4n + 1, q)

Abstract

An Erdős-Ko-Rado set of generators of a hyperbolic quadric is a set of generators which are pairwise not disjoint. In this article we classify the second largest maximal Erdős-Ko-Rado set of generators of the hyperbolic quadrics Q + (4n + 1, q), q ≥ 3.

Large {0,1,…,t}-cliques in dual polar graphs

“…, t}-cliques) of maximum size for more values of t. We provide sharp upper bounds for (d, t)-EKR sets for t ≤ 8d/5 − 2 if q ≥ 3 and for t ≤ 8d/9 − 2 if q = 2 (Theorem 4.7, Theorem 5.9, and Theorem 1.3). These results imply upper bounds on the size of the second largest example, so they might provide a reasonable basis to classify the second largest maximal (d, t)-EKR sets as it was done for EKR sets of sets [10], vector spaces [2], and some special cases in polar spaces [6,5]. Furthermore, we give non-trivial upper bounds for general t, q ≥ 3 (Theorem 8.6).…”

Equivalent definitions for (degree one) Cameron–Liebler classes of generators in finite classical polar spaces

“…We also remark that we could not generalize this classification result to other classical polar spaces, as for these polar spaces, there is not enough information known about large EKR sets in these polar spaces. For the polar spaces Q + (4n + 1, q) there are some EKR results in [8]. Since in this case, the large examples of EKR sets have much more elements than the largest known Cameron-Liebler sets, we cannot use these results.…”

Cross $t$-intersecting families for symplectic polar spaces

“…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.…”