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Statement

Setting (p. 4). The paper attributes the question to L. Moser, in connection with the Hadwiger-Nelson problem on the chromatic number of the unit-distance graph of the plane. For large $R$, let $S$ range over the measurable sets in the circle of radius $R$ such that no two points of $S$ are at distance $1$, and let $m(S)$ be the measure of $S$. The question asks to determine

$$\lim_{R\to\infty}\max m(S)/R^2. \qquad(10)$$

Conjecture (p. 4, quoted). "It seems very likely that the limit in (10) is less than $\frac14$."

The paper neither proves nor attacks the statement, and it does not discuss whether the limit exists.

Normalization

The print divides by $R^2$, not by the area $\pi R^2$ of the circle. The following is an observation of this page, not of the paper. Read literally, the statement is false. The union $A$ of the open discs of radius $1/2$ centred at the points of the lattice $2\mathbb Z^2$ has no two points at distance $1$ (two points of one disc are less than $1$ apart, and two points of different discs more than $1$ apart), and it has density $\pi/16$. Averaging over translates by the fundamental square $[0,2)^2$ shows that some translate of $A$ meets the circle of radius $R$ in measure at least $(\pi/16)\,\pi R^2$, so the maximum in (10) is at least $\pi^2/16>0.6$ for every $R$. (Croft's lower bound $0.22936$ for the largest upper density $m_1$ of such a set, which the page of Problem 232 records, is larger than $\pi/16\approx0.196$.) With $\pi R^2$ in place of $R^2$ the quotient is the proportion of the circle that such a set can fill, and the statement becomes the bound below $1/4$ on that proportion, which is how Problem 232's page reads it.

Read depth. Claims checked: the setting, display (10) and the sentence after it were read clause by clause on p. 4 of the print.

Source. P. Erdős, Problems and results in combinatorial geometry, in Discrete geometry and convexity (New York, 1982), Ann. New York Acad. Sci. 440 (1985), 1-11, Section III, p. 4. The edition read is identified on the source card.

Bears on

• Problem 232: the problem asks to estimate the largest upper density $m_1$ of a measurable plane set without two points at distance $1$, and in particular whether $m_1\le1/4$. Read with the circle's area $\pi R^2$ in place of the printed $R^2$, the conjecture asserts that the corresponding limit is below $1/4$; as printed, with $R^2$, it is false by the normalization note above. The paper proves nothing about either reading.