Large $\{0,1,\dots ,t\}$-Cliques in Dual Polar Graphs

We investigate $\{0,1,\dots ,t\}$-cliques of generators on dual polar graphs of finite classical polar spaces of rank $d$. These cliques are also known as Erd\H{o}s-Ko-Rado sets in polar spaces of generators with pairwise intersections in at most codimension $t$. Our main result is that we classify all such cliques of maximum size for $t\le \sqrt{8d\mathrm{/}5}-2$ if $q\ge 3$, and $t\le \sqrt{8d\mathrm{/}9}-2$ if $q=2$. We have the following byproducts. (a) For $q \geq 3$weprovideestimatesofHoffma{n}^{\mathrm{\prime }}sboundonthese${0, 1, \ldots, t }$-cliquesforall$t$\mathrm{.}(b)For$q \geq 3$ we determine the largest, second largest, and smallest eigenvalue of the graphs which have the generators of a polar space as vertices and where two generators are adjacent if and only if they meet in codimension at least $t+1$. Furthermore, we provide nice explicit formulas for all eigenvalues of these graphs. © We provide upper bounds on the size of the second largest maximal $\{0,1,\dots ,t\}$-cliques for some $t$.