Bounds for extremal functions on graphs

The girth conjecture proposes an asymptotically tight bound on the number of number of edges in a graph with high girth. In this project, I examine another extremal function which naturally upper bounds this quantity, and try to determine if there is a way to adapt the proof of the upper bound for this extremal function.

Most known upper bounds on the girth conjecture are constructed by examining some subgraph of the original graph, and then counting the number of vertices in the subgraph to derive bounds on the number of edges in the graph. The benefit of the proof for the other extremal function is that it provides a concrete decomposition that captures every subgraph which could have cycles of a fixed length.

As of now, I have some slides describing the problem, and some difficulties in adapting the proof.