Given $c\in (0,1)$ and a graph $G=(V,E)$ such that any subset $U\subset V$ contains an independent subset of cardinality at least $c|U|$. Does it allow to bound the chromatic number $\chi(G)$ by the constant depending only on $c$? If yes, what is the best constant?
UPDATE. For $c\geq 1/2$ we get that $G$ is bipartite. For $c<1/2$ it is proved in Gjergji's answer (by considering Kneser graph) that no uniform bound does exist.
Take the Kneser graph $K(2k+r,k)$, defined as the graph where the vertices are the $k$-element subsets of $\{1,2,\dots,2k+r\}$, and there is an edge between two vertices if the corresponding sets are disjoint. It is not hard to prove that we can take any $c < \frac{1}{2+\frac{r}{k}}$, but the chromatic number is $r+2$. So the answer to your question is negative.
Remark: The quantity $\rho(G)=\max_{H\subseteq G}\left(\frac{|H|}{\alpha(H)}\right)$ is called the Hall ratio of $G$. It is known that $\rho(G)\le \chi_f(G)\le \chi(G)$, where $\chi_f$ is the fractional chromatic number. The reason why Kneser graphs are natural to bring up in this context is because they are known as a natural example of graphs where $\chi$ is not bounded as a function of $\chi_f$.
