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Statement

Theorem 9 (p. 8), quoted: "Let $a>0$ and $\omega>0$ be real numbers. Let also $\mathbf g=D_\omega\circ R$ be an affine transformation of $\Pi$, where $R$ be a rotation and $D_\omega$ be a dilation by $\omega$. Suppose that for all $t\geqslant0$ one has

$$J_0(t)+J_0(\omega t)+J_0>-1\,,$$

where $J_0=\min_{t\geqslant0}J_0(t)=-0.4027593957\ldots$. Then for any measurable coloring of the plane $\Pi$ into two colors there is a monochromatic collinear triple $\{x,y,z\}$ such that $y=x+s$, $s\in\mathcal S_a$ and $z=x+\mathbf g(s)$. More precisely, if $R$ is a rotation by $\varphi$ then condition (15) can be replaced by

$$J_0(t)+J_0(t\omega)+J_0\bigl(t\sqrt{\omega^2-2\omega\cos\varphi+1}\bigr)>-1\,.$$

"

The first display is the paper's (15), the second its (16). The symbol $J_0$ without argument in (15) is the constant $\min_{t\geqslant0}J_0(t)$, and $\mathcal S_a$ is the circle of radius $a$ about the origin (p. 6).

Reading. The word "collinear" in the conclusion is the paper's; when $\varphi$ is not a multiple of $\pi$ the points $x$, $x+s$, $x+\mathbf g(s)$ form a nondegenerate triangle with $|xy|=a$, $|xz|=\omega a$ and angle $\varphi$ at $x$. By Lemma 8 (p. 8), $\mathbf g-I$ is a rotation followed by a dilation by $\sqrt{\omega^2-2\omega\cos\varphi+1}$, so $|yz|$ is that multiple of $a$, and the three arguments in (16) are proportional to the three side lengths. This is a filing reading, not a review verdict.

Source. I. D. Shkredov, On some problems of Euclidean Ramsey theory, arXiv:1507.02727v2 (22 July 2015), Theorem 9, p. 8; Lemma 8, p. 8; Remark 10, p. 9. The copy read is identified in the source digest.

Read depth. Claims checked: the statement, Lemma 8 and Remark 10 were read clause by clause on the page images; the proof (pp. 8--9) was read for structure only. Nothing here is independently reviewed.

Proof pointer

Pp. 8--9, following the proof of Theorem 6. The term for the pair $(x,x+s)$ gives the factor $J_0(at)$ and the term for $(x,x+\mathbf g(s))$ the factor $J_0(\omega at)$, by a change of variables through $\mathbf g^{-1}$. For the pair $(x+s,x+\mathbf g(s))$ the map $(\mathbf g-I)^{-1}$ appears, and the paper bounds that term crudely by the minimum of $J_0$, which gives the constant in (15); the paper notes that for a general transformation $\mathbf g$ the map $(\mathbf g-I)^{-1}$ does not send a circle to a circle (though it does in the collinear case of Theorem 6), and obtains (16) by applying Lemma 8. The conclusion is $(2\pi a)^{-1}(\sigma(A_*)+\sigma(B_*))\geqslant(J+J_0+1)/4>0$ with $J=\min_{t\geqslant0}(J_0(t)+J_0(\omega t))$. Not checked here.

Remark 10 (p. 9) recalls the known measurable two-coloring of the plane with no monochromatic equilateral triangle of a given side $a>0$ (the paper's [3]), and notes that for the equilateral triangle the theorem's quantity is $\min_{t\geqslant0}(2J_0(t))+J_0=3J_0=-1.208278187\ldots$, below the required $-1$. The paper adds (p. 9) that for $\omega=2$ "the minimum in (15) is greater that [sic] $-0.86$", so any triangle with two sides in ratio $1:2$ appears monochromatically.

Dependencies

• Lemma 8 (p. 8): for $\mathbf g=D_\omega\circ R$ with $R$ a rotation by $\varphi$, $\mathbf g-I=D_{\omega'}\circ R'$ with $\omega'=\sqrt{\omega^2-2\omega\cos\varphi+1}$ and $R'$ another rotation.
• Theorem 6, whose argument and notation the proof reuses.

Used by

• Theorem 1, first part.

Bears on

• Problem 173: for measurable two-colorings only, taking $\omega=|AB|/|AC|$, $\varphi=\angle BAC$ and $a=|AC|$ gives a monochromatic congruent copy of each triangle $ABC$ meeting (15) or (16). For the equilateral triangle, Remark 10 computes the quantity as $3J_0=-1.208278187\ldots$, short of the required $-1$, so the theorem does not apply to it. It says nothing about non-measurable colorings.