I will talk about an explicit construction for triangle-free graphs that do not contain a six-vertex path as an induced subgraph. Examples include the 16-vertex Clebsch graph, and the graph that arises from a complete bipartite graph by subdividing every edge of a perfect matching exactly once. We show that every connected triangle-free graph with no six-vertex induced path can be obtained from an induced subgraph of one of these two by restricted homogeneous set and homogeneous pair operations.