Ovoids and bipartite subgraphs in generalized quadrangles. (Q1889489)

A partial linear space is called a partial geometry with parameters \((s,t,\alpha)\) if (i) every line is incident with \(s+1\) points, (ii) every point is incident with \(t+1\) lines and (iii) for every line \(L\) and every point \(x\) not contained on \(L\), there are \(\alpha\) lines through \(x\) meeting \(L\). If \(\alpha=1\), then the partial geometry is called a generalized quadrangle of order \((s,t)\) (\(\text{GQ}(s,t)\)). The point graph of a partial geometry with parameters \(s,t,\alpha\) is a strongly regular graph with parameters \(v=(s+1)(1+\frac{st}{\alpha})\), \(k=s(t+1)\), \(\lambda=(s-1)+(\alpha-1)t\) and \(\mu=\alpha(t+1)\). Any strongly regular graph with such parameters is called a pseudogeometric graph. An ovoid of a pseudogeometric graph with parameters \((s,t,\alpha)\) is a coclique of size \(\frac{st}{\alpha}+1\). A \(\mu\)-subgraph of a pseudogeometric graph is the subgraph induced on the set of common neighbours of two vertices at distance 2 from each other. The authors prove the following results: (i) if a pseudogeometric graph for the generalized quadrangle \(\text{GQ}(s,s^2-s)\) contains more than two ovoids, then \(s=2\), (ii) the point graph of a \text{GQ}\((4,t)\) contains no \(K_{4,6}\)-subgraphs and (iii) if some \(\mu\)-subgraph of a pseudogeometric graph for a \text{GQ}\((4,t)\) contains a triangle, then \(t \leq 6\).