$R(x,y)$ is the largest integer such that there is an (x,y)-graph on $R(x,y) $ points

Question

im reading about numbers of Ramsey and I came across a definition that I can not understand..

Definition 1: R(x,y) is the largest integer such that there is an (x,y)-graph on R(x,y) points

the definition of the ramsey numbers is the minimum number such that there is clique $K_x$ or independent set of degree $y$. Then what is the relationship between definition 1 and the definition of Ramsey numbers

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$G$ is an $(x,y)$-graph if $x>C(G)$ and $y>I(G)$, where $I(G)$, the independence number of the graph $G$ and $C(G)$, the clique number of the graph $G$

Answer 1

First, the definition you gave,

the definition of the ramsey numbers is the minimum number such that there is no clique $K_x$ or independent set of degree $y$

is wrong. The definition of the Ramsey number $R(x,y)$ is the minimum number such that there is always either a clique of order $x$ or an independent set of order $y$ in any graph with $R(x,y)$ vertices.

A graph that does not contain either one of these is an $(x,y)$-graph in your terminology. So we could also define $R(x,y)$ as the minimum number such that no graph with $R(x,y)$ vertices is an $(x,y)$-graph.

Second, the definition you're citing,

$R(x,y)$ is the largest integer such that there is an $(x,y)$-graph on $R(x,y)$ points

is talking about the same thing, but is off by one. If $R(x,y)$ is the smallest number such that no graph on $R(x,y)$ vertices is an $(x,y)$-graph, then $R(x,y)-1$ does not have this property: there is an $(x,y)$-graph on $R(x,y)-1$ vertices. Since an $(x,y)$-graph can have $R(x,y)-1$ vertices, but not $R(x,y)$ vertices, $R(x,y)-1$ is the largest integer with this property.

There is no reason in principle why we could not have defined $R(x,y)$ as "the largest number of vertices for which some $(x,y)$-graphs still exist" as opposed to "the smallest number of vertices for which no $(x,y)$-graphs exist", because either way the problem of determining $R(x,y)$ is the same problem. But mathematicians need to agree on one convention to avoid confusion, and we chose to define $R(x,y)$ in the second way.

So this is how the definition you're citing works, and why you should forget it once you're done reading the thing you're reading.