Induced/Incomparable versus Ramsey

We consider the following problem: Let $H$ and $F$ be two graphs on $k$ vertices and assume $F \neq H$. We say that $H$ and $F$ are incomparable if neither $F$ nor $H$ contains the other. Let $H$ be a graph on $k$ vertices and let $G$ be a graph on at least $k$ vertices. Then $G$ is said to be $H$-exact if any induced subgraph of $G$ on $k$ vertices is either isomorphic to $H$ or incomparable with $H$. Exact($H$) is the family of all graphs $G$ which are $H$-exact. We pose the following problem: For a graph $H$ on $k$ vertices, determine or estimate $f(H) = \max \{n: \exists G \in \text{Exact}(H), |V (G)| = n\}$. Among the many results obtained in this paper the following are representatives concerning trees and matchings: 1. For a tree on $k \geq 3$ vertices, $(k - 1)(\left \lceil \frac{k}{2} \right \rceil -1 ) \leq f(T) \leq ( k-1)^2$. 2. For $k \geq 4$, $f(K_{1,k-1}) = (k-1)(k-2)$. 3. For $k \geq 5$, $f(P_k) = \frac{(k-1)^2}{2}$ if $k$ is odd and $f(P_k) = \frac{(k-1)(k-2)}{2}+1$ if $k$ is even. 4. $f(nK_2) = 3n$ for $n = 2, 3$ and $f(nK_2) = 4n - 4$ for $n \geq 4$.