Ovoids and Bipartite Subgraphs in Generalized Quadrangles

A point-line incidence system is called an $\alpha$-partial geometry of order $(s,t)$ if each line contains $s + 1$ points, each point lies on $t + 1$ lines, and for any point $a$ not lying on a line $L$, there exist precisely $\alpha$ lines passing through $a$ and intersecting $L$ (the notation is $pG_\alpha(s,t)$). If $\alpha = 1$, then such a geometry is called a generalized quadrangle and denoted by $GQ(s,t)$. It is established that if a pseudogeometric graph for a generalized quadrangle $GQ(s,s^2-s)$ contains more than two ovoids, then $s = 2$. It is proved that the point graph of a generalized quadrangle GQ(4,t) contains no K 4,6-subgraphs. Finally, it is shown that if some $\mu$-subgraph of a pseudogeometric graph for a generalized quadrangle $GQ(4,t)$ contains a triangle, then $t\le6$.