Let $C>0$. There exists $ϵ>0$ such that if $n$ is sufficiently large the following holds.For any $x_1,\dots ,x_n\in [-1,1]$ there exist $y_1,\dots ,y_n\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+ϵ)n$ with $P(x_i)=y_i$ for at least $(1-ϵ)n$ many $1\le i\le n$, then$$\underset{x\in [-1,1]}{\max }|P(x)|>C.$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

What you wrote is exactly the conjecture Erdős posed at the end of his 1967 paper on Lagrange interpolation. He explicitly calls it “probably” true and adds: *“I have not even proved it if $m=n$.”* ([Renyi Institute][1]) It is also listed as **open** on the Erdős Problems site. ([Erdős Problems][2])

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.