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A shorter version of this question was posted on Math Stack Exchange. Let $V$ be a nonempty set. $(V,S)$ is a graph if $S\subseteq\binom V2,$ a triple system if $S\subseteq\binom V3,$ a quadruple ...

Definition. $h(n_1,n_2)$ is the least number $m$ such that, if the edges of $K_m$ are colored with two colors, $1$ and $2,$ then for some color $i\in\{1,2\}$ there is a set $W\subseteq V(K_m)$ such ...