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Source. Section II, Concluding remarks, pp. 778--779, of Peter Frankl and Vojtech Rödl, All triangles are Ramsey, Transactions of the American Mathematical Society 297 (1986), no. 2, 777--779, doi:10.1090/S0002-9947-1986-0854099-6, as identified on the source card. Ramsey sets are defined on p. 777, as recalled on Theorem 1.

Statement

The remark is unlabeled. It states, on p. 779:

• The four points $(1,0,-2,0)$, $(0,0,1,-2)$, $(1,-2,0,0)$, $(0,1,0,-2)$ of $\mathbb R^4$ are coplanar and span a symmetric trapezoid with sides $\sqrt{10},\sqrt8,\sqrt{10},\sqrt2$ and diagonals of length $\sqrt{14}$, and this point set is Ramsey: for every $r$, if $n\ge n_0(4,2,r)$ (the Ramsey number for $2$-subsets, $4$-element sets and $r$ colors) then every $r$-coloring of $\mathbb R^n$ has a monochromatic configuration isometric to it.
• By the product theorem, infinitely many other symmetric trapezoids are Ramsey. The remark names no further trapezoid.
• The authors were unable to prove any pentagon Ramsey, and the dimension produced by the method of Stage 1 tends to infinity as the number of points of the configuration grows (pp. 778--779).
• The authors also announce that all simplices in arbitrary dimensions are Ramsey, with a proof they call less elementary deferred to a later paper; this paper gives no proof of it.

The squared distances of the four points, computed for this page, are $8$ and $2$ for the two parallel sides, $10$ for the two legs and $14$ for both diagonals, as printed.

Proof pointer

P. 779. Each pair $i<j$ in $\{1,\ldots,n\}$ is sent to the point with coordinate $1$ at $i$, $-2$ at $j$ and $0$ elsewhere; Ramsey's theorem for $2$-subsets gives four indices whose six pairs share a color, and four of those six points form the trapezoid.

Dependencies

Ramsey's theorem for $2$-subsets and the product theorem of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus (1973), both as stated on p. 777. Read depth: claims checked; the remark was read clause by clause on pp. 778--779 and the distances recomputed.

Bears on

• Problem 174: one symmetric trapezoid, and the unnamed family the product theorem builds from it, are Ramsey sets in the sense of the problem's statement. The remark decides no other four-point set and gives no characterization. The announcement on simplices is proved in Frankl and Rödl's later paper, recorded on the source card of that paper.