Polarity graphs coming from projective planes have an intimate connection with the extremal number for the 4-cycle. We will discuss finding a dense subgraph of a polarity graph and its relationship to the parameter ex(n, C4). In addition, motivated by some questions of Mubayi and Williford, we will discuss the chromatic and independence numbers of various polarity graphs. This is joint work with Sam Peng and Craig Timmons.
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