Erdős problem 1133

Erdős proved that, for any $C>0$, there exists $\epsilon>0$ such that if $n$ is sufficiently large and $m=\lfloor (1+\epsilon)n\rfloor$ then for any $x_1,\ldots,x_m\in [-1,1]$ there is a polynomial $P$ of degree $n$ such that $\lvert P(x_i)\rvert\leq 1$ for $1\leq i\leq m$ and $\max_{x\in [-1,1]}\lvert P(x)\rvert>C$. The conjectured statement would also imply this, but Erdős in [Er67] says he could not even prove it for $m=n$.