Two Consequences of The Recent Discovery That χ(R~2,1)>4.

摘要

The two consequences referred to in the title are: 1. For any coloring of the plane with red and blue such that some distance is forbidden for red, the blue set contains a translate of each 4-point set in the plane. (This strengthens a famous theorem of R. Juhász [11].) 2. There is no 2×2 generalized product coloring of the group (R~2, +, (0,0)) which forbids the distance 1. (See the paper for definitions. This result has no relation to anything famous, so far as the author know, but it is mildly surprising.)