Infinite generalized friendship graphs (Q793756)

Let \(\tau\geq 2\) and \(\lambda\geq 1\) be integers, and let c be an infinite cardinal number. It is shown that \(\tau \leq \lambda +1\) is necessary and sufficient for the existence of (simple) infinite graphs of order c such that every subset of \(\tau\) vertices has exactly \(\lambda\) common neighbors. Furthermore, whe\(n\tau \leq \lambda +1\) then for each c and for each d \((\tau +1\leq d\leq \aleph_ 0)\) there exist \(2^ c\) such pairwise non-isomorphic graphs of order c and chromatic number d. When \(\lambda\geq 2\), any such graph of order c and its complement must both be regular of degree c.