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Statement

Theorem 1 (p. 1), quoted: "Let $ABC$ be a nondegenarate [sic] triangle such that $|AB|/|AC|=\omega$. Suppose that

$$\min_{t\geqslant0}(J_0(t)+J_0(\omega t))\geqslant-0.5972406\,,$$

where $J_0$ is the zeroth Bessel function. Then any measurable coloring of $\mathbf R^2$ into two colors contains a monochromatic triangle. Further, if

$$\min_{t\geqslant0}(J_0(t)+J_0(\kappa t)+J_0((1+\kappa)t))>-1\,.$$

Then for any measurable coloring of the plane $\Pi$ into two colors there is a monochromatic collinear triple $\{x,y,z\}$ such that $y\in[x,z]$ and $\|z-y\|/\|y-x\|=\kappa$."

Here $\Pi=\mathbf R^2$, a $k$-coloring is a partition of $\Pi$ into $k$ disjoint sets (the colors), and a measurable coloring is one whose colors are measurable sets (p. 1). The paper states (p. 1) that Theorem 1 follows from Theorem 6 and Theorem 9 of Section 3.

Reading. The first part names no relation between the monochromatic triangle and $ABC$. It is read here through Theorem 9 (p. 8), applied with the dilation factor $\omega$ and the rotation by the angle $\angle BAC$: that theorem gives, for every $a>0$, a monochromatic triple $x$, $y=x+s$, $z=x+\omega R(s)$ with $\|s\|=a$, a triangle similar to $ABC$ with $|xy|=a$ in the role of $|AC|$, so with $a=|AC|$ a congruent copy. The constant $-0.5972406$ agrees to its printed digits with $-1$ minus the value $-0.4027593957\ldots$ that Theorem 9 prints for $\min_{t\geqslant0}J_0(t)$, and the first hypothesis implies Theorem 9's condition (15). The second part is Theorem 6 (p. 6), whose hypothesis is stated for all $t\geqslant0$ with $\kappa>0$, and which fixes $\|y-x\|=a$ for any prescribed $a>0$. Theorem 1 itself leaves the range of $\kappa$ unstated. These are filing readings, not review verdicts.

Source. I. D. Shkredov, On some problems of Euclidean Ramsey theory, arXiv:1507.02727v2 (22 July 2015), Theorem 1, p. 1. The copy read is identified in the source digest.

Read depth. Claims checked: the statement was read clause by clause on the page image. Nothing here is independently reviewed.

Proof pointer

No separate proof: the paper derives Theorem 1 from Theorem 9 (pp. 8--9) and Theorem 6 (pp. 6--7); see those pages.

Dependencies

• Theorem 6 (p. 6), for the collinear triple.
• Theorem 9 (p. 8), for the triangle.

Bears on

• Problem 173: for measurable two-colorings only, the first part gives a monochromatic copy of each nondegenerate triangle whose side ratio $\omega$ meets the Bessel bound (by the reading above, a congruent copy), and the second part gives the degenerate collinear triples meeting the second bound. It says nothing about non-measurable colorings or about triangles outside these conditions.