English

Reductions and necessary conditions for tall Borel Ramsey ideals

Logic 2026-07-13 v1

Abstract

An ideal I\mathcal{I} on ω\omega has the Ramsey property if I+(I+)22\mathcal{I}^{+}\to(\mathcal{I}^{+})^2_2: every 22-colouring of the pairs of an I\mathcal{I}-positive set has an I\mathcal{I}-positive homogeneous subset. Whether a tall Borel ideal can have the Ramsey property is an open question of Hru\v{s}\'ak, Meza-Alc\'antara, Th\"ummel and Uzc\'ategui; a coanalytic example exists in ZFC, so a negative answer must use definability essentially. Our main theorem, a synthesis of the results of the paper, states that a tall Borel Ramsey ideal admits no countable local reading. Below every positive set, such an ideal is not a countable intersection of topologically represented (or tall analytic PP-) ideals, and its quotient has no countable dense subset. Moreover, every quotient name for a new real has uncountable width, the colouring witnessing non-selectivity of the generic ultrafilter is never read continuously on a positive condition, and hereditary tall subfamilies saturate every finite window of barrier dimensions coherently but never all dimensions at once. We prove separately that a weakly selective q+q^+ ideal admits no positive EDfin\mathcal{ED}_{\mathrm{fin}}-carrier. The converse question -- must Borelness force a properness-like countable reading on some positive condition? -- is stated in three precise forms with proved consequences: two of them would refute tall Borel Ramsey ideals outright, the third the strictly weaker Nash--Williams class. The main question remains open.

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Cite

@article{arxiv.2607.12099,
  title  = {Reductions and necessary conditions for tall Borel Ramsey ideals},
  author = {José de Jesús Pelayo Gómez},
  journal= {arXiv preprint arXiv:2607.12099},
  year   = {2026}
}

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30 pages