This note describes two lemmas for Ramsey number R(p, q; 4), which help us to deduce lower bounds better than the corresponding results of Shastri (1990).
1. Introduction
Let S be a set. We denote by S t4) the collection of subsets of S with exactly 4 elements. We call the elements of S t4~ the 4-tuples of S. We call S t4~ the 4-tuple-class of S. A coloring of S t4~ with two colors 'red' and 'blue' is a map:
we call this a coloring of S ~4), for short, in the remainder of this note.
For every 4-tuple u we call C(u) the color of u. For a subset X of S, if all elements of X ~4) (these elements are 4-tuples of S) are colored red under the coloring C, we call X ~4) red-monochromatic; if all elements of X t4) are colored blue under coloring C, we call Xt4) red-monochromatic; if all elements of Xt4) are colored blue under coloring C, we call X ~4~ blue-monochromatic.
The Ramsey number R(p, q; 4) is the minimal integer n such that every coloring of the 4-tuple-class of the set S with cardinality n has a p-set whose 4-tuple-class is red-monochromatic or a q-set whose 4-tuple-class is blue-monochromatic For a set S with cardinality R(p, q; 4) -1, there exists a coloring for S ~4) such that the 4-tuple-class of every p-subset of S is not red-monochromatic and the 4-tuple-class