We’ll start with Greyson presenting the proof of Theorem 1.24 (a binomial coefficient upper bound on the off-diagonal Ramsey numbers).
Everyone should have a look at Proposition 1.25 (improving the Greenwood and Gleason upper bound on off-diagonal Ramsey numbers by 1, in certain circumstances).
Luca will present two arguments for a couple of small exact Ramsey numbers (Proposition 1.26 and 1.27); this outlining the proof of Proposition 1.25.
Everyone should read through propositions 1.26 and 1.27, and the discussion afterwards; I’ll make some further comments on this.
The book introduces a couple of black boxes. The first is a coloring of a graph on 13 vertices that has neither a red \(K_3\) nor a blue \(K_5\) (arrange the 23 vertices in a circle. Use a red edge to join each vertex to its neighbors immediately to the right and immediately to its left, as well as to the vertices 5 to its right and 5 to its left. All other edges are blue). We’ll talk about how to efficiently verify that this coloring does what it claims to do. The second black box is the Paley graph of order 17, which is claimed not to have a red \(K_4\) nor a blue \(K_4\). We’ll discuss verifying this, too.
Everyone should think about how a proof for Theorem 1.28 should go.
Ted will present the proof of Theorem 1.29 (a lower bound for the diagonal Ramsey numbers). Everyone should read this proof, too.