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Statement

Notation (pp. 58--59). For distinct points $x_1,\ldots,x_n$ in $E_k$, let $t$ be the number of distinct distances among them and $a_1\ge a_2\ge\cdots\ge a_t$ the multiplicities of those distances, so that $\sum_{i=1}^t a_i=\binom n2$. Then $f_k(n)$ is the largest possible value of $a_1$ and $g_k(n)$ the smallest possible value of $t$, over all choices of the $n$ points. For $k=1$ the paper notes $f_1(n)=g_1(n)=n-1$.

Display (12) and the conjecture (p. 59). As printed: "I observed in 1945 that

$$n^{1+c/\log\log n}<f_2(n)<c_2n^{3/2} \tag{12}$$

and conjectured that the lower bound in (12) is best possible or at least not far from being best possible." The paper attributes the lower bound to the triangular or square lattice, and the upper bound originally to the fact that the unit-distance graph of the points contains no $K(2,3)$.

The Erdős--Sós questions (p. 59). Erdős and V. T. Sós conjectured that the $n$ points attaining $f_2(n)$ must contain an equilateral triangle, a square, or at least four points determining at most $2$ (or perhaps $3$) distinct distances. As printed: "Further we asked: Is it true that $f_2(n)-a_2\to\infty$ as $n\to\infty$? Is it true that the configurations which maximize $a_1$ are the same which minimize $t$? The answer is almost certainly no."

Later bounds reported (p. 59). Without references, the paper reports Szemerédi's $f_2(n)=o(n^{3/2})$; Beck and Spencer's $f_2(n)<n^{3/2-\varepsilon}$ for some $\varepsilon>0$; the improvement by Fan Chung, Szemerédi, Trotter and Spencer to $f_2(n)<n^{4/3}$, as printed with no constant; and, for distinct distances, Erdős's 1946 $g_2(n)>\sqrt{n-1}-1$, L. Moser's $g_2(n)>cn^{2/3}$, Fan Chung's $g_2(n)>cn^{5/7}$ and a further $g_2(n)>cn^{3/4}$, credited to no one.

Source. P. Erdős, Extremal problems in number theory, combinatorics and geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pp. 51--70, PWN, Warsaw, 1984; MR 87a:11001; printed pp. 58 (notation) and 59 (display (12), the conjecture, the questions and the later bounds). The edition read is identified in the source digest.

Read depth. Claims checked: the notation, (12), the conjecture, the questions and the reported bounds were read clause by clause on the page images. The paper proves none of them; nothing here is independently reviewed.

Proof pointer

None printed. The paper names the constructions and the forbidden $K(2,3)$ but gives no argument.

Dependencies

None stated.

Bears on

• Problem 959: the site's [Er84d] source. The printed question is whether $f_2(n)-a_2\to\infty$. Filing observation, not a review verdict: the sentence before it concerns the configurations attaining $f_2(n)$, and the print does not say whether $a_2$ is taken in such a configuration; the site instead asks to estimate the largest gap $a_1-a_2$ over all $n$-point sets, a different quantity.
• Problem 90: the conjecture that the lower bound in (12) is best possible, read as $f_2(n)\le n^{1+O(1/\log\log n)}$, is the question of Problem 90, the largest distance multiplicity being the largest number of unit distances after scaling. The print's alternative "or at least not far from being best possible" is weaker and unquantified. Not among the site's sources for the problem.