We aim to prove that if $k \ge 3$ and $e2^{1-\binom{k}{2}}\binom{n}{k-2} \le 1$ then $R(k,k) >n$
Now I understand that we colour the edges of $K_n$ red and black with probability 1/2. For each k-set S let $A_S$ mean the event of S being monochromatic. We have $P(A_S)=2^{1-\binom{k}{2}}$
What I'm having trouble with is the maximum degree of the dependency graph. For me it says that if S and T are fixed they're related if they share an edge, $\vert S \cap T \vert \ge 2$. Which is fine, but then what I'm stuck with is the next bit:
$$d=\vert\left\{T : \vert S \cap T \vert \ge 2 \right\}\vert < {k\choose{2}}{n\choose{k-2}}$$
The upper bound on the RHS is fine by me, but isn't the "d=" bit wrong? Does it not count the case where S=T and as we know, the dependency graph should be loopless?
Many thanks Kurt