Let C>0. There exists epsilon>0 such that if n is sufficiently large the following holds. For any x_1,…,x_n∈ [-1,1] there exist y_1,…,y_n∈ [-1,1] such that, if P is a polynomial of degree m C.

Erdős Problem #1133 — Let C>0.

Erdős’s problem asks whether every sufficiently large node set in $[-1,1]$ admits bounded labels such that any polynomial of degree below $(1+\varepsilon)n$ fitting almost all labels must have arbitrarily large uniform norm. No proposer date beyond Erdős’s attribution is recorded.

Known results

Erdős, reported in 1967: for every $C>0$ , some $\varepsilon>0$ works so that, for sufficiently large $n$ , every set of $m=loor{(1+\varepsilon)n}$ nodes admits a degree- $n$ polynomial bounded by $1$ on the nodes but exceeding $C$ on $[-1,1]$ .
Erdős reported that he could not prove even the case $m=n$ .

Unverified density-principle argument

A proposed argument using Beurling’s interpolation theorem, the substitution $x=\cos\theta$ , and a block-pigeonhole principle claims to yield the robust obstruction. It has not been verified or published as a solution, and the problem remains marked open.

Current status (as of August 2026): Erdős’s weaker result is known, but the stated robust interpolation conjecture remains open and has no verified proof or counterexample.

Erdős Problem #1133 — Let $C>0$ .

Primary source

Additional references

Progress summary

Solutions 0