On Ramsey Like Theorems , Problems and Results

(1) n → (k1, . . . , k`)` means that if we split the r-tuples of a set, S, |S| = n into ` classes, than for some i, 1 ≤ i ≤ ` there is a subset Si ⊆ S, |Si| ≥ ki all of whose r-tuples are in the i-th class. Denote by F (`) r (k1, . . . , k`) the smallest n for which (1) holds. The determination of F (`) r (k1, . . . , k`) is probably hopeless and may not be a ”reasonable” problem, just as the determination of the n-th prime by a simple explicit formula is not reasonable. Very few exact results are known and all those are for r = 2 (r = 1 is trivial) [1]. It is perhaps not quite hopeless to try to get asymptotic results but even here our knowledge is meager, or to be more precise non-existent.