Ambrus 2023 density planar sets avoiding unit distances

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conjecture_p10: The paper conjectures, on numerical evidence, that the fractional chromatic number and the geometric fractional chromatic number of the planar unit-distance graph both equal 4.

corollary_1: For every measurable periodic 1-avoiding planar set A and every finite planar unit-distance graph G, 1/delta(A) >= chi_f(G), so m_1(R^2) is at most 1/chi_f(R^2).

corollary_2: For every measurable periodic 1-avoiding planar set A and every finite planar unit-distance graph G, 1/delta(A) >= chi_gf(G), the geometric fractional chromatic number, so m_1(R^2) is at most 1/chi_gf(R^2).

theorem_1: Every Lebesgue measurable planar set with no two points at distance 1 has upper density at most 0.2470, so m_1(R^2) < 1/4, as Erdős conjectured.

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Gergely Ambrus, Adrián Csiszárik, Máté Matolcsi, Dániel Varga, Pál Zsámboki, The density of planar sets avoiding unit distances. Math. Program. 207 (2024), no. 1-2, 303-327. DOI 10.1007/s10107-023-02012-9. arXiv:2207.14179.

Theorem 1 bounds the upper density of every Lebesgue measurable planar set containing no two points at distance 1 by 0.2470, so m_1(R^2) < 1/4 and Erdos's conjecture on this quantity holds. The method is a common generalization of the fractional-chromatic-number bounds from unit-distance graphs and the harmonic-analysis linear-programming approach (Szekely, de Oliveira Filho-Vallentin, Keleti-Matolcsi-de Oliveira Filho-Ruzsa, Ambrus-Matolcsi), using the complete inclusion-exclusion constraints from finite point sets of at most 24 points (the final bound comes from a 23-point set X_23) in a linear program over the autocorrelation function; Corollary 1 records the companion fact that m_1(R^2) <= 1/chi_f(G) for every finite unit-distance graph G in the plane, and Corollary 2 gives the analog with the geometric fractional chromatic number chi_gf(G) >= chi_f(G), whose linear program adds a constraint for each pair of congruent vertex subsets. As consequences the authors reprove chi_m(R^2) >= 5 and show that four colors cannot cover more than a 0.988 fraction of the plane; the paper reports Croft's 0.22936 as the best lower bound known when it was written. For problem 1070 this is the statement-cited primary source for the measurable bound m_1(R^2) <= 0.2470. Averaging translates of a dense 1-avoiding set gives f(n) >= m_1(R^2) n, so Croft's construction yields f(n) >= 0.22936n; the upper bound m_1(R^2) < 1/4 shows that this averaging argument alone cannot give f(n) >= n/4. It is a measure-theoretic density theorem, not a theorem about finite unit-distance graphs.

The copy read for this card is arXiv:2207.14179v3 (28 July 2023), at https://arxiv.org/abs/2207.14179. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2207.14179), every other right reserved.

Result pages.

• Theorem 1 (p. 4): every Lebesgue measurable 1-avoiding planar set has upper density at most 0.2470, so m_1(R^2) < 1/4; with the consequences chi_m(R^2) >= 5 and the 0.988 covering bound (p. 2).
• Corollary 1 (p. 8): for every measurable periodic 1-avoiding planar set A and every finite planar unit-distance graph G, 1/delta(A) >= chi_f(G); hence m_1(R^2) <= 1/chi_f(R^2).
• Corollary 2 (p. 10): the same with the geometric fractional chromatic number chi_gf(G) of Definition 2 (pp. 9-10); hence m_1(R^2) <= 1/chi_gf(R^2).
• Conjecture (p. 10, unnumbered): the paper conjectures chi_f(R^2) = chi_gf(R^2) = 4, on numerical evidence.

Read status. Claims checked: the statements on the result pages were read clause by clause on the printed pages of the copy named above. The proofs of the corollaries were read; the computer-assisted proof of Theorem 1 was read for structure only, and its linear program and dual certificate were not recomputed.

Bears on.

• #1070: the paper does not mention the problem. Theorem 1 gives m_1(R^2) <= 0.2470 < 1/4, so the averaging bound f(n) >= m_1(R^2) n recorded on the problem page cannot by itself reach f(n) >= n/4; it gives no bound on f(n). The conjecture on p. 10 would imply f(n) >= n/4 for every n, a positive answer to the particular question; the problem page records a pending claim that contradicts it.
• #508: the consequence chi_m(R^2) >= 5 (p. 2) concerns colourings with measurable classes and reproves Falconer's bound; it gives no bound on the chromatic number of the plane.

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