a) Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $3$-element subset of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $3$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $3$-element subsets of it are green. b) Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $k$-element subset ($k>3$) of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $k$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $k$-element subsets of it are green.