Let $n=R(p,q)$ be the Ramsey number i.e. the smallest positive integer such that is the edges of a complete graph with $n$ vertices be $2$-colored using colours $R$ and $B$ then either there exists a $K_p$ of colour $R$ or a $K_q$ of colour $B$.Now we know that $R(p,2)=p$ and $R(2,q)=q$ for all $p,q\geq 2$.Our objective is to find an upper bound for $R(p,q)$ .For that our instructor made the following claim:
$R(p,q)\leq R(p-1,q)+R(p,q-1)$
He gave it as an exercise and using that he showed that $R(m,\ell)\leq {m+\ell-2 \choose m-1}$.I am unable to prove the claim given as an exercise.Can someone show me a way out?
Addendum
I found a similar question on stack exchange An upper bound for a graph Ramsey number. But it deals with Ramsey numbers $R(G,H)$ of two graphs $G$ and $H$ ,with which I am not so comfortable.Also I did not understand the argument given there because I am not very comfortable with graph theory.