Sách Upper and lower bounds for F (4, 4; 5)

1 Introduction
In this note, we shall only consider graphs without multiple edges or loops. If G is
a graph, then the set of vertices of G is denoted by V (G), the set of edges of G by
E(G), the cardinality of V (G) by | V (G)| , and the complementary graph of G by G. The
subgraph of G induced by S ⊆ V (G) is denoted by G. A cycle of order n is denoted
by Cn. Given a positive integer n, Zn = { 0, 1, 2,


, n ư 1} , and S ⊆ { 1, 2,


, ⌊n/2⌋} ,
let G be a graph with the vertex set V (G) = Zn and the edge set E(G) = { (x, y) :
min{ | x ư y| , n ư | x ư y| } ∈ S} , then G is called a cyclic graph of order n, denoted by
Gn(S). G is an (s, t)-graph if G contains neither clique of order s nor independent set
of order t. We denote by R(s, t) the set of all (s, t)-graphs. An (s, t)-graph of order n
is called an (s, t; n)-graph. We denote by R(s, t; n) the set of all (s, t; n)-graphs. The
Ramsey number R(s, t) is defined to be the minimum number n for which R(s, t; n) is
not empty. In [3], it was proved that R(4, 3) = 9 and R(5, 3) = 14 which are useful in the
following.
For a graph G and positive integers a1, a2,


, a
r, we write G → (a1, a2,


, v
r)
v
if
every r-coloring of the vertices must result in a monochromatic ai-clique of color i for