The Ramsey characterisation by monochromatic line configurations

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Let X⊂EdX\subset\mathbb{E}^d be a set. Call XX Ramsey if for every natural number rr there exists nn such that every rr-colouring of En\mathbb{E}^n contains a monochromatic copy of XX. Write ℓm\ell_m for the relevant mm-point line configuration, and write En→(X,ℓm)\mathbb{E}^n\rightarrow(X,\ell_m) when every two-colouring of En\mathbb{E}^n contains either a monochromatic copy of XX in the first colour or a monochromatic copy of ℓm\ell_m in the second colour. The Ramsey characterisation conjecture. A set XX is Ramsey if and only if for every natural number mm there exists nn such that

En→(X,ℓm).\mathbb{E}^n\rightarrow(X,\ell_m).

The forward implication is already known from the characterisation using arbitrary fixed configurations K⊂EmK\subset\mathbb{E}^m; the conjecture asserts that line configurations alone suffice for the converse, which would give a simpler characterisation of Ramsey sets.

References

Primary source

David Conlon and Yu-Han Wu, “More on lines in Euclidean Ramsey theory”, arXiv:2208.13513 (2022).

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