Abstract
A point-line incidence system is called an α-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 α lines passing through a and intersecting L (the notation is pG α(s,t)). If α = 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 μ-subgraph of a pseudogeometric graph for a generalized quadrangle GQ(4,t) contains a triangle, then t ≤ 6.
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Makhnev, A.A., Makhnev, A.A. Ovoids and Bipartite Subgraphs in Generalized Quadrangles. Mathematical Notes 73, 829–837 (2003). https://doi.org/10.1023/A:1024053914404
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DOI: https://doi.org/10.1023/A:1024053914404