OAI.SharpRamseyFive.main

The theorem states that two asymptotic properties hold for the Ramsey number R(5,t). Here R(5,t) is the least n such that every simple graph on n labelled vertices contains either a clique of exactly 5 vertices or an independent set of exactly t vertices. The first property: there is a constant C>0 such that for every ε>0 there is a threshold t₀ with, for all t≥t₀, both t⁴/(log t)^(3+ε) ≤ R(5,t) and R(5,t) ≤ C·t⁴/(log t)³, using the natural logarithm and a real power for the exponent 3+ε. The second property: the quantity (4 log t − log R(5,t))/log log t tends to 3 as t tends to infinity. The theorem asserts the conjunction of these two statements.