Erdős Problem #1133

declared status 'open'. Formalized: no. Let $C>0$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds. For any $x_1,\ldots,x_n\in [-1,1]$ there exist $y_1,\ldots,y_n\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\epsilon)n$ many $1\leq i\leq n$, then$$\max_{x\in [-1,1]}\lvert P(x)\rvert >C.$$ Current best: Erd\H{o}s 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 Erd\H{o}s in [Er67] says he could not even prove it for $m=n$. Prize: no. Tags: analysis, polynomials.

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