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Source. The paragraph on p. 47 of P. Erdős, Combinatorial problems in geometry, Math. Chronicle 12 (1983), 35--54, the transcript of an invited address at the 17th New Zealand Mathematics Colloquium (Dunedin, 17--19 May 1982), as named on the source card. The lecture numbers none of its statements; the pages are the journal's own.

Statement

Conjecture (p. 47, from the Euclidean Ramsey papers). If $S$ is a set in the plane no two of whose points are at distance $1$, then the complement of $S$ contains four points forming the vertices of a unit square.

Theorem (p. 47, R. Juhász, as reported). If $S$ is a set in the plane containing no two points at distance $1$, then for every set of four points $x_1,\dots,x_4$ the complement of $S$ contains four points congruent to them. In particular the conjecture holds.

Reported limit (p. 47). Juhász also observed that this cannot be extended to every $k$: for large $k$ there are $S$ with no two points at distance $1$ and points $x_1,\dots,x_k$ such that the complement of $S$ contains no $y_1,\dots,y_k$ congruent to them. Erdős says it may still hold for five points.

Read depth. Claims checked: the passage was read clause by clause on the page images of the print. A second reader checked the statement, hypotheses, label and page against the print.

Proof pointer

The paper gives no proof; it says Juhász's paper appeared, Erdős thought, in the Journal of Combinatorial Theory.

Dependencies

None.

Bears on

• Problem 214: the conjecture is the problem's question, and the lecture reports Juhász's theorem, which answers it yes. The lecture is a report, not the source of the proof.