Conlon 2026 non spherical sets versus lines euclidean

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David Conlon, Jakob Führer, Non-spherical sets versus lines in Euclidean Ramsey theory. Canadian Mathematical Bulletin 69(1) (2026), 179-183. doi:10.4153/S0008439525101082. arXiv:2406.07718. The copy read for this card is arXiv v1 (11 June 2024).

Theorem 1 states that for every finite non-spherical set X there exists a natural number m such that E^n does not arrow (X, l_m) for all n, where l_m is m collinear points at consecutive distance one; this verifies a conjecture of Conlon and Wu and, granted the spherical sets conjecture, would complete their proposed characterization of Ramsey sets. The construction is explicit, built from the Erdos et al. linear relation sum c_j |x_j|^2 = B satisfied by every copy of a non-spherical X, with a rational basis for the span of the coefficients; the verification uses Weyl's equidistribution theorem and the Erdos-Turan-Koksma inequality. The introduction records the history for the simplest non-spherical set l_3: Conlon and Wu's probabilistic proof gave m at most 10^50, Führer and Toth improved it to m at most 1177 and Currier, Moore and Yip to m at most 20. Problem 188 does not fall under Theorem 1: its red configuration is a unit pair, which is spherical, and it asks for the least m in the plane. The paper bears on it only as context for the (X, l_m) program, and its m = m(X) is ineffective, since the equidistribution cutoff is not quantified.

Source: https://arxiv.org/abs/2406.07718. The arXiv record (https://arxiv.org/abs/2406.07718, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #188

Results to transcribe.

• Theorem 1: For every finite non-spherical set X there is a natural number m with E^n not arrowing (X, l_m) for all n, verifying a conjecture of Conlon and Wu; the bound on m is not made explicit.
• Construction (Section 2.1): Uses the Erdos-Graham-Montgomery-Rothschild-Spencer-Straus relation sum_j c_j |x_j|^2 = B for copies of a non-spherical X, a rational basis of the coefficient span, and an explicit coloring verified by Weyl equidistribution and the Erdos-Turan-Koksma inequality.
• Recorded bounds for X = l_3: Conlon-Wu give m at most 10^50, Führer-Toth m at most 1177, and Currier-Moore-Yip m at most 20.