Problem 750

Erdős Problem #750: declared status 'open'. Formalized: yes. Let $f(m)$ be some function such that $f(m)\to \infty$ as $m\to \infty$. Does there exist a graph $G$ of infinite chromatic number such that every subgraph on $m$ vertices contains an independent set of size at least $\frac{m}{2}-f(m)$? Prize: no. Tags: chromatic number, graph theory.