Wrt Ramsey numbers I have the following identity given to me:
$ R(m, n) \leq R(m-1, n)+R(m, n-1) $
And i have the following bases cases: $R(m,2)=m$ and $R(2,n)=n$. One has to prove that:
$R(m, n) \leq\left(\begin{array}{c}{m+n-2} \\ {m-1}\end{array}\right)$
It is obvious that we have to go on splitting the two terms on the RHS into pairs of terms decrementing the indices by 1 each time. But the appearance of the combination is non-obvious.