Infinite friendship graphs with infinite parameters (Q918996)

We study infinite graphs in which every set of \(\kappa\) vertices has exactly \(\lambda\) common neighbours. We prove that there exist \(2^{\sigma}\) such graphs of each infinite order \(\sigma\) if \(\kappa\) is finite and that for \(\kappa\) infinite there are \(2^{\lambda}\) of them of order \(\lambda\) and none of cardinality greater than \(\lambda\) (assuming the GCH). Further, we show that all a priori admissible chromatic numbers are in fact possible for such graphs.