Node sets forcing every low-degree near-interpolant to exceed a fixed bound (Erdős #1133)

Conjecture (Erdős). For every $C>0$ there exists $\epsilon>0$ such that, for all sufficiently large $n$, the following holds. Given any nodes $x_1,\ldots,x_n\in[-1,1]$, one can choose target values $y_1,\ldots,y_n\in[-1,1]$ so that every polynomial $P$ of degree $m<(1+\epsilon)n$ satisfying $P(x_i)=y_i$ for at least $(1-\epsilon)n$ of the indices $1\le i\le n$ obeys $$\max_{x\in[-1,1]}\lvert P(x)\rvert>C.$$ In words: no matter where the nodes are placed, there are bounded targets in $[-1,1]$ that cannot be matched on all but an $\epsilon$-fraction of the nodes by any polynomial of degree only slightly above $n$ without the polynomial growing larger than $C$ somewhere on $[-1,1]$.

Acceptance. FULLY RESOLVES: a complete proof of the conjecture — for each $C>0$ exhibit (or prove existence of) the required $\epsilon>0$ and threshold $n_0$ and prove the target-choice property for all $n\ge n_0$ — machine-checkable (Lean, extending the existing formal-conjectures statement) preferred, otherwise a full written proof; OR a disproof exhibiting some $C>0$, arbitrarily large $n$, and node sets $x_1,\ldots,x_n$ for which every choice of bounded targets is defeated by a degree-$<(1+\epsilon)n$ near-interpolant of sup-norm $\le C$. ADVANCES, any of: (a) prove the degree-tight case $m=n$ explicitly left open by Erdős; (b) establish the statement for a restricted class of node configurations (e.g. Chebyshev or equispaced nodes) with an explicit $\epsilon(C)$ and proof; (c) prove a quantitative strengthening giving a lower bound on the forced growth $\max_{[-1,1]}\lvert P\rvert$ in terms of $C,\epsilon,n$ rather than the qualitative $>C$. Deliver the proof (Lean artifact or written argument) or the counterexample family with certified sup-norm bounds.