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Source. The paragraph on p. 45 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

Definition (p. 45). A finite set $S$ in some Euclidean space is Ramsey if for every $k$ there is an $n_k$, depending only on $k$ and $S$, such that whenever the points of $n_k$-dimensional space are divided into $k$ classes, some class contains a set congruent to $S$. The lecture stresses congruent, not merely similar.

Reported results (p. 45), from the papers of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus:

1. The unit square is Ramsey; for two colours, fifteen points in $5$-dimensional space suffice, in that every $2$-colouring of them has a monochromatic unit square.
2. Every brick (rectangular parallelepiped), in any number of dimensions, is Ramsey.
3. Every Ramsey set lies on a sphere.

Erdős says these are the only general theorems known.

Question (p. 45). Is the isosceles triangle with one angle $120^\circ$ and the others $30^\circ$ Ramsey: is it true that for $n_k$ large enough there is a finite set in $n_k$-dimensional space such that every division of it into $k$ classes has a class containing a triangle congruent to this one? Erdős calls it the simplest unsolved problem of the subject.

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 proves none of these; it cites the three joint papers, one in the Journal of Combinatorial Theory (1973) and two in the proceedings of the Keszthely meeting.

Dependencies

None.

Bears on

• Problem 174: background. The problem asks for a characterization of the Ramsey sets; the lecture reports the necessary condition (lying on a sphere), two sufficient classes (the unit square, the bricks) and an open triangle case, and proves nothing new.