TITLE: Ramsey number $R(3,7)=23$
QUESTION [5 upvotes]: Im reading the prove the Ramsey of number $R(3,7)=23$ of "Some Graph Theoretic Results Associated with Ramsey’s Theorem"JACK and JAMES YACKEL.
$G$ is an $(x,y)$-graph if $x>C(G)$ and $y>I(G)$, where $I(G)$ is the maximum number of points of $G$ that can be chosen so that no two are joined by an edge and $C(G)$ is the maximum number of points in any complete subgraph of $G$. (in other word, $G$ is a graph that does not have cliques of size $x$ neither independent sets of size $y$)
Define: Let $G$ be a graph and $p$ a point of $G$. By $H_1$ we will mean
the graph spanned by all points of $G$ which are joined to $p$ by an edge.
$H_2$ will denote the graph spanned by all points different from $p$ which are
not joined to $p$ by an edge. In the proof to Lemma $2$ we proved that if
$G$ is an $(x, y)$-graph then $H_1$ is an $(x- 1, y)$-graph and $H_2$ is an $(x, y- 1)$-graph. We will indicate this disection of $G$ by stating that $p$ is the preferred.
point.
The autor define: $R'(x,y)$ is the largest integer such that there is an
$(x, y)$-graph on $R'(x,y)$ points. In other words $R'(x,y)=R(x,y)-1$, at the end of the article prove that $R(3,7)\geq 22$, 
but there are several things that I can't understand,

If someone could break me down a bit, the demonstration would greatly appreciate it.
I don't understand why there are many triangles and they are in H2..
In $R'(3,7)\leq 22$ prove that $G$ contains only points of valence $6$.

REPLY [4 votes]: Addendum - A proof of the claim that the reduced $H_2$ has no independent set of size 6
Suppose that there is a set $X$ of 6 independent points.
First we prove $X$ is not independent in $H_2$
In view of the symmetry of $H_2$ we can suppose that $p_{12} \in X$.
The only points in $H_2$ independent of $p_{12}$ are
$p_{13}-p_{24},p_{14}-p_{23},p_{15}-p_{26},p_{16}-p_{25}$
where the dashes indicate some of the edges in $H_2$. We can only have one of each pair of joined points in $X$ and so there can be at most $5$ independent points in $H_2$.
We now know that $X$ contains two adjacent points in the Hamiltonian path
There are 15 possibilities (unless we establish some symmetries amongst the points in the path). We shall consider one possibility; that $X$ contains $p_{35}$ and $p_{16}$.
The only points independent of both $p_{35}$ and $p_{16}$ in the reduced $H_2$ are either adjacent to one of them in the Hamiltonian path or contain both one of the suffices $3,5$ and one of the suffices $1,6$. So the possibilities are $p_{12}-p_{36},p_{13}-p_{56},p_{15}-p_{23}$
where the dashes indicate some of the edges in the reduced $H_2$. Again there can be at most $5$ independent points.