Usually, a "triangle" in graph theory is a complete graph consisting of three vertices (denoted by " K3 "), see the definition in [Diestel] p. 3 or the definition in [Bollobas] p. 5. This corresponds to the definition of a "triangle graph" (which is a more precise term) in Wikipedia "Triangle graph", https://en.wikipedia.org/wiki/Triangle_graph, 27-Jul-2025: "In the mathematical field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle. The triangle graph is also known as the cycle graph C3 and the complete graph K3."
Often, however, the term "triangle" is also used to denote a corresponding subgraph of a given graph ("triangle in a graph"), see, for example, Wikipedia "Triangle-free graph", 28-Jul-2025, https://en.wikipedia.org/wiki/Triangle-free_graph: "In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges."
In this subsection, a triangle (in a graph) is defined as a set of three vertices of a given graph. In this meaning, a triangle with ) is neither a graph nor a subgraph, but it induces a triangle graph as subgraph of the given graph .
We require that there are three (different) edges connecting the three (different) vertices of the triangle. Therefore, it is not sufficient for arbitrary hypergraphs to say "a triangle is a set of three (different) vertices connected with each other (by edges)", because there might be only one or two multiedges fulfilling this statement. We do not regard such degenerate cases as "triangle".
The definition df-grtri is designed for a special purpose, namely to provide a criterion for two graphs being not isomorphic (see grimgrtri). For other purposes, a more general definition might be useful, e.g., ComplSubGr for complete subgraphs of a given size (proposed by TA). With such a definition, we would have ComplSubGr (at least for simple graphs), and the definition df-grtri may become obsolete.