Abstract:

There is a long history of using subgraph counting identities in Ramsey theory. When McKay and Radziszowski worked on new upper bounds for $R(5,5)$ and $R(4,6)$ they used computers to find patterns in the number of induced subgraphs. They found a new kind of pattern and conjectured a subgraph counting identity. I will give a proof of their conjecture using an algebraic bookkeeping method developed for subgraph counting.