Timeline for Monochromatic triangles in every two-coloring of the plane?
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 14, 2016 at 4:50 | comment | added | Gerry Myerson | The conjecture that the equilateral triangle is the only exception appears in Erdős, P.; Graham, R. L.; Montgomery, P.; Rothschild, B. L.; Spencer, J.; Straus, E. G., Euclidean Ramsey theorems. I, J. Combinatorial Theory Ser. A 14 (1973), 341–363, MR0316277 (47 #4825). | |
| Sep 10, 2010 at 21:53 | comment | added | David Eppstein | More generally it is known that, if T is any right triangle, then every 2-coloring of the plane contains a monochromatic copy of T. Soifer ("The Mathematical Coloring Book", Problem 40.10, p.491) credits it to Leslie Shader ("All right triangles are Ramsey in E^2!", JCTA 20 (1976), pp. 385-389, MR0409260). | |
| Sep 10, 2010 at 20:15 | comment | added | itamar | Hey, Can anyone advice how to prove that for every two-coloring of the plain there will exist a triangle with with side lengths 1, √3 ,2 ? Thanx Itamar | |
| Jul 30, 2010 at 20:51 | vote | accept | Matthew Kahle | ||
| Jul 30, 2010 at 14:03 | history | edited | Tony Huynh | CC BY-SA 2.5 |
fixed hyperlink
|
| Jul 29, 2010 at 21:32 | comment | added | Matthew Kahle | The De Bruijn–Erdős Theorem says that the chromatic number of an infinite graph (if it exists) is the maximum chromatic number of its finite subgraphs. Shelah and Soifer gave examples to suggest that the chromatic number of the plane might depend on set theoretic axioms. Here is an example of a coloring problem in the plane (with countably many colors), where the answer is equivalent to the Continuum Hypothesis: mathoverflow.net/questions/273/… | |
| Jul 29, 2010 at 19:55 | comment | added | Eric Tressler | @James: tinyurl.com/2f3ftzw is a link to an excerpt from Ramsey Theory on the Integers by Landman and Robertson, stating a (not fully general) version of the compactness principle. | |
| Jul 29, 2010 at 19:49 | comment | added | Eric Tressler | @James: Yes. There is a general compactness principle involved. See Graham, Rothschild, and Spencer's book Ramsey Theory, introduction. For all of these problems, including Hadwiger-Nelson, this is the case. I.e., if the chromatic number of the plane is 5, then there exists a finite unit distance graph that forces this. Similar statements hold for other problems of this type. | |
| Jul 29, 2010 at 19:40 | comment | added | James Martin | Hearing this question I would be interested to know the following. If, for a given T, the answer is "yes" (i.e. if in any 2-coloring of the plane there exists a monochromatic copy of T) is this necessarily for a "finite reason"; i.e., must there then exist a finite 3-uniform hypergraph which can be embedded in the plane such that all its "edges" are triangles congruent to T, which is not 2-colorable? | |
| Jul 29, 2010 at 19:40 | answer | added | Eric Tressler | timeline score: 22 | |
| Jul 29, 2010 at 19:20 | history | asked | Matthew Kahle | CC BY-SA 2.5 |