Triply existentially complete triangle-free graphs

A triangle-free graph G is called k-existentially complete if for every induced k-vertex subgraph H of G, every extension of H to a (k + 1)-vertex triangle-free graph can be realized by adding another vertex of G to H. Cherlin [11,12] asked whether k-existentially complete triangle-free graphs exist for every k. Here, we present known and new constructions of 3-existentially complete triangle-free graphs.