Lower bounds for some Ramsey numbers

AbstractLet n, r, u1, u2,…,uk be positive integers satisfying ui ⩾ r for i = 1,2,…,k. The symbol n → (u1, u2,…,uk)r means that, for any partition of the r-subsets of an n-set S into k classes C1, C2,…,Ck, there is a ui-subset of S all of whose r-subsets belong to Ci for some i, 1 ⩽ i ⩽ k. A theorem of F.P. Ramsey asserts that, if r, u1, u2,…,uk are given, then n → (u1, u2,…,uk)r for all sufficiently large n. n ↦ (u1, u2,…,uk)r denotes the negation of n → (u1, u2,…,uk)r. In this paper a number of results of the form n ↦ (u1, u2,…,uk)3 are obtained