Exoo 2018 chromatic number plane is at least

../

claim_2_1: Exoo and Ismailescu's Claim 2.1: a configuration of 79 points with coordinates in Q[sqrt 3, sqrt 11, sqrt 247] such that every proper 4-coloring of the plane gives two of its points at distance sqrt(11/3) the same color.

claim_3_1: Exoo and Ismailescu's Claim 3.1: a configuration of 49 points in Q[sqrt 3, sqrt 11]^2 containing two fixed points P and Q at distance sqrt(11/3) such that every proper 4-coloring either colors P and Q differently or has a monochromatic equilateral triangle of side 1/sqrt 3.

claim_4_1: Exoo and Ismailescu's Claim 4.1: for any equilateral triangle ABC of side 1/sqrt 3 there is a unit distance graph of order 627 containing A, B and C that has no proper 4-coloring giving A, B and C the same color.

main_theorem: Exoo and Ismailescu's unnumbered main result: no proper 4-coloring of the plane exists, so chi(E^2) >= 5, proved by chaining three finite configurations and giving a different proof of de Grey's bound.

---

Geoffrey Exoo, Dan Ismailescu, The chromatic number of the plane is at least 5 - a new proof. arXiv preprint (2018). arXiv:1805.00157. The copy read for this card is arXiv:1805.00157v1 (1 May 2018). The arXiv record names arXiv's non-exclusive distribution license (arXiv:1805.00157), every other right reserved.

The paper reproves de Grey's result χ(E²) ≥ 5 by a different chain of three finite configurations. Claim 2.1 (p. 2) exhibits 79 points, with coordinates in Q[√3,√11,√247]², forcing in any proper 4-coloring a monochromatic pair at distance √(11/3); Claim 3.1 (p. 5) gives a 49-point configuration in Q[√3,√11]² turning such a monochromatic √(11/3) pair into a monochromatic equilateral triangle of side 1/√3; Claim 4.1 (p. 6) gives a unit-distance graph of order 627 containing that triangle which cannot be 4-colored when its three vertices share a color, built up from a 51-vertex unit-distance graph. Combining the three assertions contradicts the existence of a proper 4-coloring, so χ(E²) ≥ 5 (p. 2). The method is explicit coordinate construction, all graphs but one having embeddings with vertices of the form ((a√3+b√11)/36, (c+d√3√11)/36) with integer a,b,c,d, plus computer verification of the coloring properties of the finite graphs; data files are posted online. Section 5 (p. 11) assembles the three claims into one explicit 5-chromatic unit-distance graph, of order much larger than de Grey's first graph.

Source: https://arxiv.org/abs/1805.00157.

Bears on. #508: the main theorem gives the lower bound χ(E²) ≥ 5 by a proof different from de Grey's; it reproves that known bound, does not improve it, and gives no upper bound.

Results. Labels and pages are those of v1.

• Main theorem (p. 2, unnumbered): no proper 4-coloring of the plane exists, so χ(E²) ≥ 5.
• Claim 2.1 (p. 2): 79 points with coordinates in Q[√3,√11,√247]² force, in any proper 4-coloring, two identically colored points at distance √(11/3).
• Claim 3.1 (p. 5): 49 points in Q[√3,√11]² containing a fixed pair P, Q at distance √(11/3) such that in any proper 4-coloring either P and Q differ in color or a monochromatic equilateral triangle of side 1/√3 exists.
• Claim 4.1 (p. 6): for any equilateral triangle ABC of side 1/√3 there is a unit-distance graph of order 627 having A, B, C among its vertices and no proper 4-coloring that gives A, B and C one color.

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