The triangle-free Krein graph Kre(r) is strongly regular with parameters (Formula presented.). The existence of such graphs is known only for r = 1 (the complement of the Clebsch graph) and r = 2 (the Higman–Sims graph). A.L. Gavrilyuk and A.A. Makhnev proved that the graph Kre3 does not exist. Later Makhnev proved that the graph Kre4 does not exist. The graph Kre(r) is the only strongly regular triangle-free graph in which the antineighborhood of a vertex (Formula presented.) is strongly regular. The graph (Formula presented.) has parameters (Formula presented.). This work clarifies Makhnev’s result on graphs in which the neighborhoods of vertices are strongly regular graphs without 3-cocliques. As a consequence, it is proved that the graph Kre(r) exists if and only if the graph (Formula presented.) exists and is the complement of the block graph of a quasi-symmetric 2-design.