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Statement

Notation (p. 969). $(x_i,x_j)$ is the distance between $x_i$ and $x_j$.

Theorem 3 (p. 970, stated without proof). Let $x_1,\dots,x_n$ be $n$ points in four-dimensional space. There are an absolute constant $c$ and an $n_0=n_0(\varepsilon,c)$ such that for $n>n_0(\varepsilon,c)$ any $\frac14n^2(1+\varepsilon)$ of the distances $(x_i,x_j)$ include more than $n^c$ distinct numbers.

The print does not quantify $\varepsilon$; the dependence of $n_0$ on it reads it as an arbitrary positive number. Unlike Theorem 2, the statement does not say the points are distinct.

Proof pointer

None in the paper. Erdős says (p. 971) that the proof is similar to that of Theorem 2, and that both are no doubt special cases of a more general theorem estimating the number of distinct values among $\frac14n^2+nf(n)$ of the distances.

Read depth

Claims checked: the statement was read on the page images of the print. The paper gives no proof, so none was checked.

Dependencies

None.

Source. P. Erdős, On some applications of graph theory to geometry, Canad. J. Math. 19 (1967), 968--971; the edition read is named on the source card.

Bears on

No problem page of this corpus.