Inequalities and asymptotic bounds for Ramsey numbers

AbstractPartitions of pairs of elements of a set into classes are considered. The maximal sizes of sets which admit partitions of the pairs into classes satisfying the conditions of Ramsey are considered and several inequalities are obtained which relate these maximal sizes for various Ramsey conditions. The principal result concerns the partitions into two classes and the upper bound R(N + 1, N + 1) ⩽ C∗log log Nlog N2NN is obtained