Independence, induced subgraphs, and domination in $K_{1,r}$-free graphs

Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define ${\alpha }_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with \emph{chromatic number} at most~$k$, we prove that if $G$ is $K_{1,r}$-free, then ${\alpha }_{\mathcal{F}}(G)\le (r-1)k\gamma (G)$, where $\gamma (G)$ is the \emph{domination number}. When $\mathcal{F}$ is the family of empty graphs, this bound simplifies to $\alpha (G)\le 2\gamma (G)$ for $K_{1,3}$-free (claw-free) graphs, where $\alpha (G)$ is the \emph{independence number} of $G$. For $d$-regular graphs, this is further refined to the bound $\alpha (G)\le 2\left(\frac{d+1}{d+2}\right)\gamma (G)$, which is tight for $d\in \{2,3,4\}$. Using Ramsey theory, we extend this framework to edge-hereditary graph families, showing that for $K_{1,r}$-free graphs, we have ${\alpha }_{\mathcal{F}}(G)\le r(K_r,{\mathcal{F}}^{\ast })\gamma (G)$, where ${\mathcal{F}}^{\ast }$ is the set of graphs not in $\mathcal{F}$. Specializing to $K_q$-free graphs, we show ${\alpha }_{\mathcal{F}}(G)\le (r(K_q,K_r)-1)\gamma (G)$. Finally, for the \emph{$k$-independence number} ${\alpha }_k(G)$, we prove that if $G$ is $K_{1,r}$-free with order $n$ and minimum degree $\delta \ge k+1$, $${\alpha }_k(G)\le \left(\frac{(r-1)(k+1)}{\delta -k+(r-1)(k+1)}\right)n,$$ and this bound is sharp for all parameters.