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Source. Unnumbered conjecture, p. 72, of P. Erdős, "Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems," Mathematica (Cluj) 10 (33) (1968), 65-73; see the source card.

Statement

Conjecture (p. 72, unnumbered; introduced with "Probably the following result also holds"). For every $A$, however large, there is an $\varepsilon>0$ such that if $n>n_0(A,\varepsilon)$, then for every $-1\le x_1<\cdots<x_n\le1$ there are $y_1,\ldots,y_n$ with $|y_i|\le1$, $i=1,\ldots,n$, such that every polynomial $P_m(x)$ of degree $m<(1+\varepsilon)n$ with $P_m(x_i)=y_i$ for at least $n(1-\varepsilon)$ values of $i$ satisfies

$$\max_{-1\le x\le1}|P_m(x)|>A.$$

Erdős writes that the result, if true, clearly contains Theorem 4, and that he has not proved it even if $m=n$.

Read depth. Read clause by clause on the printed page.

Bears on

• Problem 1133: source. The conjecture is the problem's assertion, with the problem's $C$ in place of $A$; the paper poses it and proves no case of it.