The maximum number of edges in a graph with fixed edge-degree (Q1382818)

For \(t\geq 17\) and \(n\geq 2t+2\), let \(G\) be a graph with \(n\) vertices whose complement is connected, and such that for all non-adjacent vertices \(u,v\), there are at least \(t\) common neighbours. It is proved that \(| E(G)|\geq \lceil((2t+1)n- 2t^2- 3)/2\rceil\) for \(n\leq 3t-1\), and \(| E(G)|\geq (t+1)n- t^2- t-3\) for \(n\geq 3t\). Furthermore, it is shown that both of these results are sharp.