Actualización : ahora se conoce el conjunto de obstrucciones (es decir, la "barrera" NxM entre los tamaños de cuadrícula colorable e incolorable) para todos los 4 colores sin rectángulo monocromático .
¿Alguien tiene ganas de probar 5 colores? ;)
La siguiente pregunta surge de la teoría de Ramsey .
Considere una coloración del gráfico de cuadrícula n -by- m . A existe siempre que cuatro celdas con el mismo color estén dispuestas como las esquinas de algún rectángulo. Por ejemplo, ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) , y ( 1 , 0 ) formar un rectángulo monocromática si tienen el mismo color. Del mismo modo, ( 2 , 2 ) , ( 2 , 6 ) (monochromatic rectangle
and form a monochromatic rectangle, if colored with the same color.
Question: Does there exist a -coloring of the -by- grid graph that does not contain a monochromatic rectangle? If so, provide the explicit coloring.
Some known facts:
- -by- is -colorable without a monochromatic rectangle, but the known coloring scheme does not appear to extend to the -by- case. (I'm omitting the known -by- coloring because it would very likely be a red herring for deciding -by-.)
- -by- is NOT -colorable without a monochromatic rectangle.
- -by- and -by- are also unknown cases; an answer to these would be interesting as well.
Disclaimer: Bill Gasarch has a $289 (USD) bounty on a positive answer to this question; you can reach him through his blog. A note on etiquette: I'll make sure he knows the source of any correct answer (should one arise).
He brought it up again during a rump session at Barriers II, and I find it interesting, so I'm forwarding the question here (without his knowledge; though I highly doubt he would mind).