Shkredov 2015 problems euclidean ramsey theory

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corollary_4: For every sufficiently large prime p congruent to -1 mod 4 and every non-equilateral triangle ABC in the plane over F_p, every two-coloring of that plane contains a monochromatic triangle congruent to ABC.

corollary_7: For every real a > 0, every measurable two-coloring of the plane contains a monochromatic collinear triple x, y, z with y between x and z and |y - x| = |z - y| = a.

theorem_1: Shkredov's main theorem: a measurable two-coloring of the plane contains a monochromatic triangle when a side ratio omega of a nondegenerate triangle satisfies a Bessel-function bound, and a monochromatic collinear triple with step ratio kappa when a second Bessel-function bound holds.

theorem_3: For every sufficiently large prime p and every invertible affine map g of the plane over F_p with g - I invertible, every two-coloring of the plane and every nonzero a give a monochromatic triple x, x + s, x + g(s) with s on the sphere of radius a.

theorem_6: For real a > 0 and kappa > 0 with J_0(t) + J_0(kappa t) + J_0((1 + kappa)t) greater than -1 for all t >= 0, every measurable two-coloring of the plane has a monochromatic collinear triple x, y, z with y between x and z, |y - x| = a and |z - y| = kappa a.

theorem_9: For real a > 0 and omega > 0 and g a rotation followed by a dilation by omega, a Bessel-function condition gives in every measurable two-coloring of the plane a monochromatic triple x, x + s, x + g(s) with |s| = a, which yields monochromatic copies of triangles with two sides in ratio omega.

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I. D. Shkredov, On some problems of Euclidean Ramsey theory. arXiv preprint (2015). arXiv:1507.02727. The copy read for this card is arXiv:1507.02727v2 (22 July 2015). The arXiv record names arXiv's non-exclusive distribution license (arXiv:1507.02727), every other right reserved.

Shkredov studies the measurable version of the Euclidean Ramsey question asking for a monochromatic non-equilateral triangle in any two-coloring of the plane. Theorem 1 (p. 1), assembled from Theorem 9 (p. 8) and Theorem 6 (p. 6) of Section 3, gives two criteria in terms of the zeroth Bessel function: if a nondegenerate triangle ABC has side ratio omega = |AB|/|AC| with min over t >= 0 of J_0(t) + J_0(omega t) at least -0.5972406, then every measurable two-coloring of R^2 contains a monochromatic triangle, which Theorem 9 supplies as a copy of ABC at every scale; and if min over t >= 0 of J_0(t) + J_0(kappa t) + J_0((1 + kappa) t) > -1, then every measurable two-colouring contains a monochromatic collinear triple x, y, z with y between x and z and |z - y|/|y - x| = kappa (Theorem 6 gives |y - x| = a and |z - y| = kappa a for every a > 0). The constant -0.5972406 agrees to its printed digits with -1 minus the value -0.4027593957... that Theorem 9 prints for the minimum of J_0, so the first condition implies Theorem 9's condition (15). Corollary 7 (p. 7) applies Theorem 6 with kappa = 1: every measurable two-coloring contains a monochromatic three-term progression x, y, z with |y - x| = |z - y| = a, for each a > 0. The proofs use elementary Fourier analysis on the plane and depend essentially on there being only two colors; Section 2 develops a model finite-field analog over F_p x F_p with a slightly stronger conclusion. Theorem 3 (p. 3) is the finite-field result: for large p, an invertible affine g with g - I invertible and any a != 0, every two-coloring of F_p x F_p has a monochromatic triple x, x + s, x + g(s) with s on the sphere of radius a. For problem 173 the paper gives partial results only: they hold for measurable colorings and for triangles meeting the Bessel conditions. Later work (Currier, Moore and Yip, currier_2024) removes the measurability hypothesis for the equal-step three-term progression.

Source: https://arxiv.org/abs/1507.02727.

Bears on. #173: for measurable two-colorings of the plane only, Theorem 9 (p. 8) gives a monochromatic congruent copy of each triangle whose side ratio and angle meet its Bessel condition (15) or (16), and Theorem 6 (p. 6) and Corollary 7 (p. 7) give monochromatic collinear triples with steps a and kappa a, including the equal-step case. For the equilateral triangle, Remark 10 (p. 9) computes Theorem 9's quantity as 3 J_0 = -1.208278187..., short of the required -1, so Theorem 9 does not apply to it; a measurable two-coloring with no monochromatic equilateral triangle of a given side is known. The paper says nothing about non-measurable colorings; Theorem 3 and Corollary 4 are analogs over F_p x F_p, not statements about the plane.

Results.

• Theorem 1 (p. 1): the paper's main theorem, the two Bessel-function criteria, derived from Theorems 6 and 9.
• Theorem 3 (p. 3): monochromatic triples x, x + s, x + g(s) in every two-coloring of F_p x F_p, for large p; Corollary 5 (p. 5) is recorded on the same page.
• Corollary 4 (p. 4): for large p = -1 (mod 4), a monochromatic triangle congruent to any non-equilateral triangle in F_p x F_p; the proof (p. 5) says it leaves some restrictions in one case unstated.
• Theorem 6 (p. 6): monochromatic collinear triples with steps a and kappa a under condition (11), for every a > 0.
• Corollary 7 (p. 7): monochromatic collinear triples with equal steps a, for every a > 0.
• Theorem 9 (p. 8): monochromatic triples x, x + s, x + g(s) for a rotation-dilation g under condition (15) or (16), with Lemma 8 (p. 8) and Remark 10 (p. 9).

Read status: claims checked for the statements above, each read clause by clause on the page images; the proofs were read for structure only, and the numerical step in the proof of Corollary 7 was not repeated. Nothing is independently reviewed.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.