  There are finitely many graphs with diameter $2$ and girth 5. What if the
girth 5 assumption is relaxed? Apart from stars, are there finitely many
triangle-free graphs with diameter $2$ and no $K_{2,3}$ subgraph? This question
is related to the existence of triangle-free strongly regular graphs, but
allowing for a range of co-degrees gives the question a more extremal flavour.
More generally, for fixed $s$ and $t$, are there infinitely many twin-free
triangle-free $K_{s,t}$-free graphs with diameter 2? This paper presents
partial results regarding these questions, including computational results,
potential Cayley-graph and probabilistic constructions.
