The chromatic number of two families of generalized Kneser graphs related to finite generalized quadrangles and finite projective 3-spaces (Q2039997)

Summary: Let \(\Gamma\) be the graph whose vertices are the chambers of the finite projective space \(\text{PG}(3,q)\) with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is \((q^2+q+1)(q+1)^2\). For \(q\geqslant 43\) we determine the largest independent set of \(\Gamma\) and show that every maximal independent set that is not a largest one has at most constant times \(q^3\) elements. For \(q\geqslant 47\), this information is then used to show that \(\Gamma\) has chromatic number \(q^2+q\). Furthermore, for many families of generalized quadrangles we prove similar results for the graph that is built in the same way on the chambers of the generalized quadrangle.