Euclidean Ramsey Theory: Lines Problem

(La,Lb;n) means for all 2-col of R^n exists red unit-a-AP or blue unit-b-AP

(La,Xb;n) means for all 2-col of R^n exists red unit-a-AP or blue X-config, and X size b

Outline

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Paper on Measureable Colorings

Paper on Measurable Colorings

Intro

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(L2,Lb)

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Szlam. Yes (L2,X3;2)s. 1999

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Juhasz. Yes (L2,X4;2). 1979

Tsaturian. Yes (L2,L5;2). 2017

Arman-Tsaturian. Yes (L2,L6;3). 2017

Csizmadia-Togh. exists X NOT (L2,X8;2). 1994

Arman-Tsaturian. Yes (L2,L(n+3);n). 2017

Currie-Mody-Xie-Zhang. No (L2,L6330};R^2). No on something. 2026

Conlon-Fox. Yes (L2,L(2^{cn};R^n). No on something. 2018

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(L3,Lb)

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Currier-Moore-Yip. No (L3,L20;R^n), (L4,L18;R^n)), (L5,L10;R^n). 2024

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Fuhrer-Toth. No (L3,L1177;R^n). 2024

Conlon-Wu. No (L3,L10^50;R^n). 2022

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(L4,Lb), (L5,Lb), (L6,Lb)

Currier-Moore-Yip. No (L3,L20;R^n), (L4,L18;R^n)), (L5,L10;R^n). 2024

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Erdos et al. NOT: (L6,L6;2). 1973

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Papers on chi(En): Asymptotic

FW paper, exp lower bound on chi(En)

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Raigorodskii paper- slight improvement on FW

LR Paper on exp upper bound on chi(Rn)

Prosanov. A new proof of the LR up bd for chi(En) by Prosanov

Prosanov. chromatic number of spheres

Papers on chi(En): Concrete

Upper Bds on chi(En) in low dim by Arman-Bondarenko-Prymak-Radchenko

RT. chi(E3) LE 15

Chrekaskhin. On the Chrom numb of small-dim E. Spaces

Coulson. 15-coloring of of E3

Coulson. 18-coloring of of E3

Exoo. chi(En) for small n

Exoo. chi(E4)

Kahle. chi(Ed) for d=8,9,10,11,12

Oren. chi(E3) GE 6