Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Statement
The first question (Section 7, p. 173), raised by G. Fejes Tóth and Erdős during the meeting: "Can one find a finite set of unit intervals in the unit square, no two of which intersect and which are maximal with respect to this property?" Here unit intervals are segments of length , and maximal means that no further unit segment in the square avoids all of them.
The answer reported (pp. 173-174). Danzer found a simple example, drawn as Fig. 3 (p. 174), for which Erdős paid a prize. Another participant of the meeting, whom Erdős does not name, found a second example, Fig. 4 (p. 174), in which the extension of the upper side of the lower left quadrangle passes through the lower right vertex of the square. Erdős notes that in both examples the segments' positions can be varied. The paper gives the figures without a written proof of maximality.
Further questions (pp. 173-174). Erdős says it is not clear what happens when the unit square is replaced by other regions, or in the unit square when two of the intervals may share an endpoint and nothing else. He then asks (p. 174): "Let be any region and let there be given in a maximal set of disjoint unit intervals. Can such a set ever be denumerable?"
Source. P. Erdős, Some combinatorial and metric problems in geometry, Intuitive geometry (Siófok, 1985), Colloq. Math. Soc. János Bolyai 48, North-Holland, Amsterdam-New York, 1987, 167--177 (MR 89i:52012); Section 7, printed pp. 173-174, with Figs. 3 and 4 on p. 174.
Read depth. Claims checked: both questions, the report of the two examples and the description of Fig. 4 were read clause by clause on the page images of pp. 173-174. The maximality of the drawn families was not checked from the figures here.
Proof pointer
None in this paper; the examples are given as figures only.
Dependencies
None.
Bears on
- Problem 1071: the first question is the site's first question, which Danzer's Fig. 3 and the unnamed participant's Fig. 4 answer yes as Erdős reports; the denumerable question is the site's second question, which the site phrases as whether some region has a countably infinite maximal family. The problem page records the claims for both.