Lines in Euclidean Ramsey theory
Let ℓm be a sequence of m points on a line with consecutive points of distance one. For every natural number n, we prove the existence of a red/blue-coloring of En containing no red copy of ℓ2 and no blue copy of ℓm for any m≥2cn . This is best possible up to the constant c in the expo...
| Príomhchruthaitheoirí: | , |
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| Formáid: | Journal article |
| Foilsithe / Cruthaithe: |
Springer US
2018
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| Achoimre: | Let ℓm be a sequence of m points on a line with consecutive points of distance one. For every natural number n, we prove the existence of a red/blue-coloring of En containing no red copy of ℓ2 and no blue copy of ℓm for any m≥2cn . This is best possible up to the constant c in the exponent. It also answers a question of Erdős et al. (J Comb Theory Ser A 14:341–363, 1973). They asked if, for every natural number n, there is a set K⊂E1 and a red/blue-coloring of En containing no red copy of ℓ2 and no blue copy of K. |
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