Number of edges required to guarantee $K_3$ as subgraph
Hint: Matrix multiplication runs in time $O(n^{2.71})$. A relevant matrix is the adjacency matrix of the graph.

Actually, this has been improved several times in the past very. Recently, Le Gall improved this to time $O(n^{2.3728639})$. It is conjectured that matrix multiplication runs in time $O(n^{2+\epsilon})$ for every $\epsilon &gt; 0$.