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Source. Theorem 27 with its proof and the paragraph before it, p. 582, of P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus, Euclidean Ramsey Theorems, III, Infinite and Finite Sets (Keszthely 1973), Colloq. Math. Soc. János Bolyai 10, North-Holland (1975), 559--583, as identified on the source card.

Statement

Theorem 27 (p. 582). If $R(1,1,x)$ holds for some transcendental number $x<2$, then there exists an interval $I$ containing $x$ in its interior such that $R(1,1,y)$ holds for all $y\in I$.

The paper notes before it (p. 582) that all its monochromatic results concern triangles with an algebraic dependence among the three sides, that it has no $(a,a,b)$-triangle with $R(a,a,b)$ and $a/b$ transcendental, and that any such result "would constitute an important advance".

Proof pointer

P. 582. By compactness (from Part I) some finite planar set $S(x)$ forces a monochromatic $(1,1,x)$-triple in every two-coloring; its coordinates may be taken algebraic over $\mathbb Q(x)$, hence algebraic functions of a variable $X$ evaluated at $X=x$, and specializing $X$ to any $y$ in an interval between the nearest branch points gives a set $S(y)$ that does the same for $(1,1,y)$.

Read depth. Claims checked: the statement was read on p. 582; the proof was read for its structure only.

Bears on

• Problem 173: a conditional statement. A single transcendental isosceles triangle proved Ramsey would give a whole interval of isosceles triangles none of which is the exceptional triangle of any coloring. The paper proves the hypothesis for no $x$.