A hole is chordless cycle that length of the cycle is four or more. 


In [this post](https://mathoverflow.net/questions/99887/help-on-the-following-extremal-problem) I asked: What is the maximum number of holes that a simple graph on n vertices can have? 


Gil Kalai answered that there is no polynomial upper bound.


I need a polynomial upper bound for number of holes over following class of graphs: 
Graphs constructed from triangles such that no two triangles have more than one vertex in common. 
this graphs are not necessarily chordal, and may have holes. 


May I hope for a polynomial upper bound for number of holes of such graphs? 



Is there a polynomial upper bound for number of holes over following class of graphs?
Gil Kalai's example has no triangles - the smallest cycle has length 4. So glue a triangle to each edge of his graph. (Add a vertex, and connect it to both vertices of the edge). This graph is then constructed of triangles, but no two triangles share an edge. (and it's simple, so no two triangles share two points) None of the holes have gained chords.


Thus, it is still a counterexample.


